Transcript Slide 1

To find the trig values of any acute angle we
rotate a point about a circle as shown below:
Now we redefine the the trig functions as
y
x
y
follows: sin   , cos   , tan   ,
r
r
x
r
r
x
csc   , sec   , cot  
y
x
y
Since both x and y are positive in the first
quadrant, all the trig function values are 
in first quadrant.
What happens if the point is in the second
quadrant? Third quadrant? Fourth quadrant?
Consider a point on the
 x, y 
circle in quadrant II
y
sin   (positive)
r
x
cos   (negative)
r
r
y
x (neg)
y
tan   (negative)
x
Only sine (& csc) are positive,
all other trig values negative.
Consider a point on the circle
in quadrant III.
y
sin   (neg)
r
x
cos   (neg)
r
y
tan   (pos)
x
x (neg)
y (neg)

r
Only tangent & cotangent positive in Q III,
all other trig values are negative!
Consider a point on the circle
in quadrant IV.
y
sin   (neg)
r
x
cos = (pos)
r
y
tan   (neg)
x

x (pos)
r
Only cosine & secant positive in Q IV,
all other trig values are negative!
y (neg)
So, if we know what quadrant the terminal
side lies in, we can tell what the sign   or  
of the trig function is:
sine & cosecant 
All 
tan & cot 
cos & sec 
Q I All are positive
Q II only sin (&csc) 
Q III only tan (& cot) 
Q IV only cos (& sec) 
We can use the " All Studs Take Calculus"
to quickly determine the sign of a trig function!
Only sin & csc  in Q II
S
A
T
C
Only tan & cot  in Q III
All  in Q I
Only cos & sec  in Q IV
Find the exact trig function value of:
cos 2250.
First sketch the angle:
This forms a right triangle
in quadrant III, a 450  450  900
 whose sides are: 1,1, 2.
x 1
2
So cos 225  

r
2
2
0
1
2
or cos 225   cos 45  

2
2
0
0
0
Note: 45 is called the reference angle.
Find the exact trig function value of:
cot  2400  .
r 2
First sketch the angle: y  3
reference angle  600 , this forms a
300  600  900 triangle whose sides are:
x  1
1, 2 , 3
x 1
3
So cot  240   

y
3
3
0
1
3
or cos  240    cot 60  

3
3
0
0
Find the exact trig function value of:
sec  750

First sketch the angle:
0
sec  7500   sec  300 
 sec  3300 
reference angle  300 , this forms a
300  600  900 triangle whose sides are:
x  3 , r  2 , y  1
r
2 2 3
sec  750   sec  330   sec  30   

x
3
3
0
0
0