Transcript 7.4 Trigonometric Functions of General Angles What if the angle is not acute?

7.4
Trigonometric Functions
of General Angles
What if the angle is not acute?
Let  be any angle in standard position, and
let a , b denote the coordinates of any point,
except the origin (0, 0), on the terminal side
of  . If r  a  b denotes the distance from
(0, 0) to (a , b), then the six trigonometric
functionsof  are defined as the ratios
sin   b r
cos  a r
tan   b a
csc  r b
sec  r a
cot   a b
2
2
provided no denominator equals 0.
y

r
(a, b)
x
Find the exact value of each of the six
trigonometric functions of a positive angle 
if (-2, 3) is a point on the terminal side.
y
(-2, 3)

x
a  2, b  3
r  a 2  b2  ( 2) 2  32  13
b
3 3 13
sin   

r
13 13
r
13
csc  
b
3
a  2  2 13
r
13
cos  

sec   
a
2
r
13
13
b 3
3
tan  

a 2
2
a
2
cot    
b
3
y
r 1
P= (1, 0)
P= (a, b)
x
b 0
sin 0  sin 0    0
r 1
a 1
cos 0  cos 0    1
r 1
b 0
tan 0  tan 0    0
a 1
r 1
csc 0  csc 0  
b 0
r 1
sec 0  sec 0    1
a 1
a 1
cot 0  cot 0  
b 0
sin
y
cos
2

tan
P= (0,1)


x
2

csc
sec
cot
2

2

2

2
 sin 90
 cos 90
 tan 90
 csc 90
 sec 90
 cot 90
b 1
  1
r 1
a 0
  0
r 1
b 1
 
a 0
r 1
  1
b 1
r 1
 
a 0
a 0
  0
b 1
180 ( radians)
270 (3 2 radians)
sin 
0
1
cos
tan 
1
0
0
Not defined
csc
Not defined
1
sec
1
Not defined
cot 
Not defined
0
y
a < 0, b > 0, r > 0
a > 0, b > 0, r > 0

r
x
(a, b)
a < 0, b < 0, r > 0
a > 0, b < 0, r > 0
y
II  ,  
sin   0, csc  0
All others negative
III  ,  
tan   0, cot   0
All others negative
I (+, +)
All positive
IV  ,  
x
cos  0, sec  0
All others negative
y
II  ,  
I (+, +)
Students
All
( Sin )
III  ,  
(All functions)
IV  ,  
Take
Care
( Tangent )
( Cosine )
x
Two angles in standard position
are said to be co-terminal if they
have the same terminal side.
y

x

Let  denote a non-acute angle that
lies in a quadrant. The acute angle
formed by the terminal side of  and
either the positive x-axis or the
negative x-axis is called the
reference angle for  .
Reference
Angle

Finding the reference angle 
1. Add / subtract multiples of 360
2 
until you obtain an angle  between
0 and 360
0 and 2 radians.
2. Determine the quadrant in which the
terminal side of the angle formed by
the angle  lies.
y
  180  
or
  
 
x
    180
or
 
  360

or
 2  
Theorem: Reference Angles
If  is an angle that lies in a quadrant and if
 is its reference angle, then
cos   cos
sin   sin 
csc   csc 
tan   tan 
cot    cot 
sec   sec 
where the + or  sign depends on the
quadrant in which  lies.
Find the exact value of each of the
following trigonometric functions using
reference angles:
16
(a) cos 570
(b) tan
3
(a) 570  360  210  
 in Quadrant III, so cos < 0
  210  180  30
3
cos 210   cos 30  
2
16
16 6 10
 2 


b
3
3
3
3
10 6 4


3
3
3
 is in Quadrant III, so tan > 0
4


 
3
3
16

3
tan
 tan 
3
3 2