Study Guide for Chapter 6
1. Given that tan( )  6 and csc( )  0 , then  is in Quadrant ______. Determine the value of the other five
trigonometric functions.
2. Write cos( ) in terms of sin( ) given that  is in quadrant II.
3. Verify the identities:
(a)
sin( x) tan( x)  sin( x)
 cos( x)
tan( x)  tan 2 ( x)
(b)
(c)
cos( x)  sec( x)
  sin 2 ( x)
sec( x)
(d) cos2 ( x) tan 2 ( x)  1  cos 2 ( x)
4. Given ( )
sec( x) csc( x)

 tan( x)
sin( x) sec( x)
( ) which of the following are valid inputs? Circle your answers.
,
,
,
5. Suppose
(
. Find the exact value of
).
6. Determine the exact value of each of the following:
(
(a)
)
(b)
(c) cos2 (22.5 )  sin 2 (22.5 )
(
(e)
7. Given that
(
(a)
8. Given that
(
(a)
)
( )
and
(
)
(b)
and
( )
(b)
( )
( )
(
and
)
are obtuse angles with
)
)
(d)
are acute angles with
)
(
(
determine:
(c)
and
)
(
)
determine:
(c)
(
)
9. Using the given right triangle, find the exact values of the following. Show work.
1
5
θ
sin θ = ____________
sin 2θ = ____________
cos θ = ____________
cos 2θ = ____________
10. Use the diagram below to evaluate cos( ) where     45
5° 𝜃
7
4
11. State the domain and range of the inverse tangent function f t   arctant  .
12. Evaluate each of the following exactly. Give angles in radians.
 1 
=
 3

(b) arcsin  
(a) arctan 




 4  
 =
 3 
 6 
 5 
(i) cot  arctan 
 11  
=
 6 
(h) csc  arcsin    =



(f) arccos  cos 



  
 =
 3 
(e) arccos  cos 
(g) arccos  sin 

(c) arctan  3 
 6 
 7 
(d) cos  arccos   

1 
=
2



13. Draw a triangle for each of the following expressions. Write each expression. Assume

 5 
 =
 13  


 x 
(c) cot  arccos    =
 9 

 15  
 =
 12  
.

 9 
 2 


 1 
(d) sin  arcsec 
=
 2x  

(b) sec  arctan     =
(a) tan  arcsin 
14. Solve each equation. State (a) the principal root, (b) all solutions in the interval
( )
( )
(a)
(b) √
√
), and (c) all real roots.
15. Find all solutions in the interval [0, 2 ) .
(a)
(c)
3 cos( x) tan(2 x)  cos( x)  0
( )
( )
√
(b) 4sin( x)cos( x)  2 3 sin( x)  2cos( x)  3  0
(d)
(
)
16. Find all solutions.
(a) csc( x)   2
(c) 6sin 2 ( x) cot 2 ( x)  3
1 
x  4 3
2 
(d) cot( x)csc( x)  2cot( x)  csc( x)  2  0
(b) 8sin 