Slide 1

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 2

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 3

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 4

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 5

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 6

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 7

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 8

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 9

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 10

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 11

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 12

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 13

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 14

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 15

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 16

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 17

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 18

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 19

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 20

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 21

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 22

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 23

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 24

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 25

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 26

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 27

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 28

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 29

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 30

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 31

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 32

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 33

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 34

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 35

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 36

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 37

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 38

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 39

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 40

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 41

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 42

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 43

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 44

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 45

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 46

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 47

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 48

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 49

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 50

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 51

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 52

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53


Slide 53

Sections 8.1, 8.2, 8.3, 21.10

Trigonometric Functions of Any Angle
&
Polar Coordinates

1

Definitions of Trig Functions of Any Angle
(Sect 8.1)

Definitions of Trigonometric Functions of Any Angle
Let  be an angle in standard position with (x, y) a point on the
terminal side of  and r  x 2  y 2
y

sin  
cos  
tan  

y

csc  

r

r

y

x

r

(x, y)

r

x

r

y

x

x

sec  
cot  


x

y
2

The Signs of the Trig Functions

Since the radius is always positive (r > 0), the signs
of the trig functions are dependent upon the signs of
x and y. Therefore, we can determine the sign of
the functions by knowing the quadrant in which the
terminal side of the angle lies.

3

The Signs of the Trig Functions

4

To determine where each trig function is POSITIVE:

“All Students Take Calculus”
Translation:

S
T

A
C

A = All 3 functions are positive in Quad 1
S= Sine function is positive in Quad 2
T= Tangent function is positive in Quad 3
C= Cosine function is positive in Quad 4

*In Quad 2, sine is positive, but cosine and tangent are negative; in Quad 3, tangent is
positive, but sine and cosine are negative; in Quad 4, cosine is positive but sine and
tangent are negative.

**Reciprocal functions have the same sign. So cosecant is positive wherever sine is
positive, secant is positive wherever cosine is positive, and cotangent is positive
5
wherever tangent is positive.

Example
Determine if the following functions are positive or negative:
1) sin 210°

2) cos 320°

3) cot (-135°)

4) csc 500°

5) tan 315°

6

Examples
For the given values, determine the quadrant(s) in which
the terminal side of θ lies.
1) sin   0.3614

2) tan    2.553

3) cos    0.866

7

Examples
Determine the quadrant in which the terminal side of θ lies,
subject to both given conditions.
1) sin   0, cos   0

2) sec   0, cot  0

8

Examples
Find the exact value of the six trigonometric functions of θ if
the terminal side of θ passes through point (3, -5).

9

Reference Angles (Sect 8.2)
The values of the trig functions for non-acute angles can be found
using the values of the corresponding reference angles.

Definition of Reference Angle
Let  be an angle in standard position. Its reference
angle  ref is the acute angle formed by the terminal
side of  and the horizontal axis.
Quadrant II

 ref 

Quadrant III

 ref 

Quadrant IV

 ref 

10

Example
Find the reference angle for   2 2 5 

Solution

y

By sketching  in standard position,
we see that it is a 3rd quadrant angle.
To find  ref , you would subtract 180°
from 225 °.

 ref 


 ref

x

 ref 
11

So what’s so great about reference angles?
Well…to find the value of the trig function of any non-acute angle, we
just need to find the trig function of the reference angle and then
determine whether it is positive or negative, depending upon the
quadrant in which the angle lies.

For example,

sin 225    (sin 45  )  

1
2

In Quad 3, sin is negative

45° is the ref angle

12

Example
Give the exact value of the trig function (without using a calculator).

cos 150

13

Examples (Text p 239 #6)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

6 a ) tan 91 

b ) sec 345 

14

Examples (Text p 239 #8)
Express the given trigonometric function in terms of the same
function of a positive acute angle.

8 a ) cos 190 

b ) cot 290 

15

Now, of course you can simply use the calculator to find
the value of the trig function of any angle and it will
correctly return the (approximate) answer with the
correct sign.
Remember:


Make sure the Mode setting is set to the correct form of the
angle: Radian or Degree



To find the trig functions of the reciprocal functions (csc, sec,
and cot), use the  button or enter [original function] .

16

Example
Evaluate co t 3 2 4 .0  . Round appropriately.


Set Mode to Degree



Enter: 

OR



A N S : cot 324.0    1.38

17

HOWEVER, it is very important to know how to use the
reference angle when we are using the inverse trig
functions on the calculator to find the angle because the
calculator may not directly give you the angle you want.
Example:
Find the value of  to the nearest 0.01°

y


-12

x
r

-5

(-12, -5)

18

Examples
F in d  fo r 0     3 6 0 
1) sin    0.418

19

Examples
F in d  fo r 0     3 6 0 
2) tan   1.058

20

Examples
F in d  fo r 0     3 6 0 
3) cos    0.85

21

Examples
F in d  fo r 0     3 6 0 
4) cot    0.012, sin   0

22

BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

2 sin   1  0

23

SUPER DUPER BONUS PROBLEM
Find  for 0     360  w ithout using a calculato r.

4(sin  )  9   6
2

24

Trig functions of Quadrantal Angles
To find the sine, cosine, tangent, etc. of angles whose terminal
side falls on one of the axes (...,  180 ,  90 , 0 , 90 , 180 , 270 , 360 , ...),
we will use the circle.

(0, 1)

90

Unit Circle:


Center (0, 0)



radius = 1



x2 + y2 = 1

(-1, 0)

(1, 0)

180

0

270

(0, -1)

25

Now using the definitions of the trig functions with r = 1,
we have:

sin  
cos  

y

 y

csc  

r



1

r

1

y

y

x

x

r

1

r
tan  



y

y
x



x

sec  

1

x
cot  



x

x
y

26

Example
Find the value of the six trig functions for    9 0 

(0, 1)

sin   90   

270

(-1, 0)

(1, 0)

180



90

(0, -1)

0

cos   90   
tan   90   

y



y

r

1

x

x



r
y




1


x

csc   90   

r
y

y

sec   90   

r

1

cot   90   

x




x

1




x


y
27

Example
Find the value of the six trig functions for   0 
sin  0    y 
co s  0    x 
tan  0   

y



x

csc  0   

1

sec  0   

1

co t  0   

x



y


x


y
28

Example
Find the value of the six trig functions for 

 540 

sin  540   



cos  540   



tan  540   



csc  540   



sec  540   



cot  540   


29

Radian Measure (Sect 8.3)
A second way to measure angles is in radians.

Definition of Radian:
One radian is the measure of a central angle  that intercepts
arc s equal in length to the radius r of the circle.
In general, for  in
radians,

 

s

r

30

Radian Measure

2 radians corresponds to 360 

2  6.28

 radians corresponds to 180 

  3.14





2

radians corresponds to 90 

 1.57

2

31

Radian Measure

32

Conversions Between Degrees and Radians

1.

2.

To convert degrees to radians, multiply degrees by

To convert radians to degrees, multiply radians by


180
180



Example
Convert from degrees to radians: 210º
210 
33

Conversions Between Degrees and Radians
Example
a) Convert from radians to degrees:
 3

 3
4



4

b) Convert from radians to degrees: 3.8

(to nearest 0.1°)

3 .8 

34

Conversions Between Degrees and Radians

c) Convert from degrees to radians (exact):  6 7 5 
675 

d) Convert from radians to degrees (exact):
13

13
6



6

35

Conversions Between Degrees and Radians
Again!
e) Convert from degrees to radians (to 3 decimal places): 5 2 
52 

f) Convert from radians to degrees (to nearest tenth): 1 rad
1

36

Examples
Find  to 4 sig digits for 0    2 

sin   0.9540
Hint: There are two answers. Do you remember why?

37

38

Polar Coordinates (Sect 21.10)
A point P in the polar coordinate system is represented by an
ordered pair  r ,   .
• If r  0 , then r is the distance of the point from the pole.

•  is an angle (in degrees or radians) formed by the polar
axis and a ray from the pole through the point.

39

Polar Coordinates
If r  0 , then the point is located r units on the ray that
extends in the opposite direction of the terminal side of .

40

Example

 
Plot the point P with polar coordinates  2,
 4


.


41

Example
Plot the point with polar coordinates

  
 4, 
 3 

4

42

Plotting Points Using Polar Coordinates

a)

 5 
 3,

3 


b)

 

 2,  
4


43

Plotting Points Using Polar Coordinates
c)

 3, 0 

d)

 

  5, 
2


44

A)

B)

C)

D)

45

To find the rectangular coordinates for a point given its
polar coordinates, we can use the trig functions.
Example

  
 4, 
 3 

46

47

Likewise, we can find the polar coordinates if we are given the
rectangular coordinates using the trig functions.
Example:
Find the polar coordinates (r, θ) for the point for which the rectangular
coordinates are (5, 4). Express r and θ (in radians) to three sig digits.

(5, 4)

48

Conversion from Rectangular Coordinates to Polar Coordinates
If P is a point with rectangular coordinates (x, y), the polar coordinates
(r, ) of P are given by

r 

x  y
2

2

 ref  tan

1

y
x

You need to consider the quadrant in
which P lies in order to find the value of .
P

49

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

a)

 2,  2 

50

Find polar coordinates of a point whose rectangular coordinates
are given. Give exact answers with θ in degrees.

b)

  1,  3 

51

The TI-84 calculator has handy conversion features built-in.
Check out the ANGLE menu.

5: Returns value of r given rectangular coordinates (x, y)
6: Returns value of  given rectangular coordinates (x, y)

7: Returns value of x given polar coordinates (r, )
8: Returns value of y given polar coordinates (r, )

Check the MODE for the appropriate setting for angle measure (degrees vs. radians).
52

End of Section

53