Download Section 6.3

Section 6.3
Properties of the
Trigonometric Functions
If sin  > 0 and cos  < 0, name the quadrant in which the
angle  lies.
For sin  > 0 the y value must be positive so the angle must
be in quadrant I or II.
For cos  < 0 the x value must be negative so the angle must
be in quadrant II or III.
Therefore, this angle must lie in quadrant II.
10
3 10
and cos  
, find the value of each of the four remaining
10
10
trigonometric functions of  .
Given sin  
10
sin 
10 10
1
10
tan  




cos  3 10
10 3 10 3
10
1
1
10
csc  


 10
sin 
10
10
10
1
1
cot  
 3
tan  1
3
1
1
10
10
sec  



cos  3 10 3 10
3
10
Find the exact value of each expression. Do not use a calculator.
1
2
(a)

cos
35
2
csc 35
cos


3
(b)
 cot

3
sin
3
1
2
2
2
(a)

cos
35


sin
35


cos
35  1
2
csc 35
cos




3
(b)
 cot  cot  cot  0

3
3
3
sin
3
2
Given that sin   and cos   , find the exact value of each
5
of the remaining five trigonometric functions of  .
2
Since sin   0 and cos  0, is in quadrant II.
5
The circle has a radius of 5 so its equation is x2  y 2  52
r 5
2
2
2
csc   =
P is a point on the circle so 5  x  2
y
x 2  21 x   21
P(x,2)
r=5
θ
x  21
cos   =
r
5
2
r
5
5 21
sec  =

x  21
21
y
2
2 21
tan   =

x  21
21
x
21
cot   = 
y
2
2
Given that sin   and  is an acute angle, find the exact value of each
5
of the remaining five trigonometric functions of  .
sin   cos   1
2
2
2
2
2

co
s
 1
 
5
4 21
2
cos   1 

25 25
cos   
21
21

25
5
Since θ is in quadrant II, x
values are negative
2
sin 
2  5 
2 21
5
tan  

  


cos 
21
21 5 
21 

5
1
1 5
1
21
csc  
 
cot  

sin  2 2
tan 
2
5
sec  
1

cos 
1
5
5 21


21
21
21

5
1
Given that cot  and sin   , find the exact value of each
3
of the remaining five trigonometric functions of  .
1
Since cot   0 and sin   0, is in quadrant III.
3
2
2
2
The circle has a radius of 3 so its equation is x  y  3
r
10
2
csc


=

P is a point on the circle so r  1  9  10
y
r  10
θ
P(-1,-3)
3
r
10
sec   =
  10
x
1
y 3
tan   =
3
x 1
y
3
3 10
x
1
10 sin   =

cos   =

r
10
10
r
10
10
1
Given that cot  and sin   , find the exact value of each
3
of the remaining five trigonometric functions of  .
1
1
3
3 10
sin  



cot   1  csc 
csc 
10
10
10

2
3
1
1
1
2
tan  
 3
   1  csc 
cot  1
3
3
1
10
3 10
2

csc    1 
sin 
3 10 1
10
10
9
9
cos  


 
tan 
3
10 3
10
10
10
csc  

1
1
10
sec  


  10
9
3
cos 
10
10

Since sin θ < 0, csc θ is negative.
10
2
2
Find the exact value of:
(a) cos (60°)
(b) sin (390°)
(c)
 37 
tan  

 4 
1
(a) cos  60   cos 60 
2
1
(b) sin  390    sin 390   sin  30  360    sin 30  
2
 37
(c) tan  
 4

 37


tan



 4

  36 


tan

 

4 

4



  tan   9    tan  1
4
4
