Trigonometry Section 5.1
In 1 – 4, find sin s .
1.
1
cot s   , s in quadrant II
3
3.
tan s  
7
, sec s  0
2
2.
cos s 
4.
sec s 
5
, tan s  0
5
11
, tan s  0
4
Use the fundamental identities to find the remaining five trigonometric functions of  .
6.
1
cos    , sin   0
4
1
tan    ,  in quadrant IV
4
8.
5
csc    ,  in quadrant III
2
4
, sin   0
3
10.
sec  
5.
sin  
7.
9.
cot  
2
,  in quadrant II
3
4
, sin   0
3
Use the fundamental identities to simplify the expression. There is more than one correct form of
each answer
11.
cot  sin 
14.
1  cos 1  sec  
16.
sec  
19.
1  tan 2 
1  cot 2 
sin 
tan 
12.
cot 2  1  tan 2  
15.
17.
sin  tan   cos 
sin   csc   sin  
tan  cot 
sec 
20.
13.
18.


cos     sec 
2


sin 2   tan 2   cos 2 
For each trigonometric expression in Column I, choose the expression from Column II that
completes a fundamental identity.
Column I
Column II
21.
cos x
sin x
a)
sin 2 x
cos 2 x
22.
cos x 
b)
sin 2 x  cos 2 x
23.
1
c)
sin x
24.
sec 2 x  1
d)
cos x
25.
1  cos x  csc x
e)
cot x
2
Trigonometry Section 5.2
Perform the indicated operations and simplify the result.
1.
cot xtan x  sin x 
2.
1  sin t 2  cos 2 t
3.
1
1

1  cos x 1  cos x
Factor each trigonometric expression.
4.
sin 2   1
5.
4 tan 2   tan   3
6.
cos 4   2 cos 2   1
7.
cot 4   3 cot 2   2
Each expression simplifies to a constant, a single circular function, or a power of a circular
function. Use the fundamental identities to simplify each expression.
8.
sin  tan 
cos 
9.
sec 2   1
10.
sec cos
Verify each trigonometric identity.
11.
1  sin 2 
 cos 
cos 
12.
13.
cot x  tan x  sec x csc x
14.
cos  sin 

 sec 2   tan 2 
sec  csc 
15.
sin 4   cos 4   2 sin 2   1
16.
tan 2 x sin 2 x  tan 2 x  cos 2 x  1
17.
1
1

 2 sec 2 
1  sin  1  sin 
18.
cos   1
cos 

2
tan 
sec   1
19.
1  cos x
2
 cot x  csc x 
1  cos x
20.
1
 sec   tan 
sec   tan 
21.
sin 2  sec 2   sin 2  csc 2   sec 2 
22.
tan 2 x  1 tan x  cot x

sec 2 x
tan x  cot x
23.
sin x
sin x cos x

 csc x 1  cos 2 x 
1  cos x 1  cos x
24.
sin 4 x  cos 4 x
1
sin 2 x  cos 2 x
25.
1  sin x  cos x 2  21  sin x 1  cos x 
cos 2  tan 2   1  1
Trigonometry Section 5.3
Use the sum and difference identities for cosine to find the exact value. (Do not use a
calculator.)
  
cos  
 12 
1.
cos 75
2.
cos105
4.
 7 
cos

 12 
5.
cos 10 cos 35  sin  10sin 35
6.
cos
3.
2

2

cos  sin
sin
5
10
5
10
Write each of the following in terms of the cofunction of a complementary angle.
7.
tan 87
10.
sin
5
8

8.
cos
11.
cot 176.9814
Use the cofunction identities to find an angle
12

9.
csc 1424
12.
sec 14642
that makes each statement true.
13.
tan   cot45  2 
14.
sin   cos2  10
15.


sec   csc   20 
2

16.
sin 3  15  cos  25
Use the identities for the cosine of a sum or a difference to reduce each expression to a single
function of  .
17.
Find
20.
21.
22.
cos90   
cos0   
18.
19.
cos270   
coss  t  and coss  t .
1
3
and sin t  , s and t in quadrant II
5
5
2
1
sin s  and sin t   , s in quadrant II and t in quadrant IV
3
3
8
3
cos s  
and cos t   , s and t in quadrant III
17
5
cos s  
Verify the identity.

 cos

 sin
cos
25.
cos 2 x  cos 2 x  sin 2 x
3
12
cos

23.
4

12
sin
26.

4
24.
cos 70 cos 20  sin 70 sin 20  0
cos  x  y   cos  x  y   2cos x cos y
Trigonometry
Section 5.4
Use the identities of this section to find the exact value of each of the following.
1.
sin 15
4.
 7 
sin  

 12 
6.
tan 80  tan  55
1  tan 80 tan  55
8.
cos195
2.
tan 105
3.
tan

12
5.
sin 76 cos 31  cos 76 sin 31
7.
5

 tan
12
4
5

1  tan
tan
12
4
tan
9.
sin 165
10.
tan 255
Use the identities of this section and the previous one to express each of the following as an
expression involving functions of x or  .
11.
cos30   
12.
 3

cos
 x
 4

13.


tan   x 
4

14.


sin   x 
4

15.
sin 270   
16.
tan 360   
For each of the following, find
and the quadrant of s  t .
sin s  t , sin s  t , tan s  t , tan s  t , the quadrant of s  t ,
2
1
and sin t   , s in quadrant II and t in quadrant IV
3
3
17.
sin s 
18.
cos s  
19.
4
12
sin s   and cos t 
, s in quadrant III and t in quadrant IV
5
13
20.
cos s 
8
3
and cos t   , s and t in quadrant III
17
5
11
2
and cos t 
, s and t in quadrant IV
5
6
Verify that each statement is an identity.
21.
sin 2x  2 sin x cos x
22.
sin 210  x   cos120  x   0
23.
cos   
 tan   cot 
cos  sin 
24.
tan      tan 
 tan 
1  tan     tan 
Trigonometry
Section 5.5
Use the identities in this section to find values of the six trigonometric functions for each of the
following.
3
and  terminates in quadrant III
4
1.
 , given cos 2 
2.
x , given cos 2 x  
3.
2 x , given tan x  2 and cos x  0
5

 x 
and
12
2
2 , given sin   
4.
5
and cos   0
7
Use an identity to write each expression as a single trigonometric function or as a single number.
5.
cos 2 15  sin 2 15
6.

2 sin
3
cos
2 tan

7.
3

3
1  tan 2
8.
1
1  2 sin 2 22 
2
9.
1
2 cos 2 67   1
2
11.
1 1 2
 sin 47.1
4 2
12.
sin 2
2
2
 cos 2
5
5
 
tan 2  
 3
15.
10.
cos 2

8
cos 260
14.
 11 
sin 2 

 2 
Verify each identity.
16.
sin 
 cos    sin 2  1
18.
cos 2 
20.
tan   45  tan   45  2 tan 2
22.


cot  tan      sin     cos     cos 2 
2


2
2  sec 2 
sec 2 
sec 2 x  sec 4 x
2  sec 2 x  sec 4 x
17.
sec 2 x 
19.
1  cos 2 x
 cot x
sin 2 x
21.
tan x  cot x  2 csc 2x
Express each function as a trigonometric function of x.
23.
cos 3x
24,
tan 4 x
25.
sin 5x
3

Find the exact value of each of the following.
13.

1
2
Trigonometry
Section 5.6
Use the half-angle identities of this section to find the exact value.

1.
sin 15
2.
cos
4.
tan 195
5.
cos165
8
3.
tan 67.5
6.
sin 67.5
Find each of the following.

, given cos 

1

, with 0   
4
2
7.
cos
8.
sin
9.
tan
10.
cot
11.
cos x , given cos 2 x  
12.
sin x , given cos 2 x 
2

5

 
, given cos    , with
2
8
2

2

2
, given tan

7
, with 180    270
3
5
, with 90    180
2
, given tan   
5

 x 
and
12
2
2
3
and   x 
3
2
Use an identity to write each expression as a single trigonometric function.
1  cos 76
2
13.
16.

1  cos 147
1  cos 147
14.
1  cos 18 x
2
17.

1  cos 5 A
1  cos 5 A
1  cos 59.74
sin 59.74
15.

18.
1  cos
Verify that the equation is an identity.
19.
x 1  cos x 
cot

2
sin 2 x
21.
tan
2
2

2
 csc   cot 
20.
2
x
 tan 2  1
1  cos x
2
22.
1  tan 2

2

2 cos 
1  cos 
2
3
5
Trigonometry
Section 5.7
Express as a sum or a difference.
1.
sin 7t sin 3t
4.
4 sin 60 sin 150
2.
6 sin

4
cos

3.
4
cos 6u cos 4u 
3 cos x sin 2 x
5.
Express as a product.
6.
sin 6  sin 2
7.
cos 5x  cos 3x
8.
sin 3u  sin 7u
9.
sin 60  sin 30
10.
cos  2   cos 
11.
cos
14.
tan 4 x
3

 cos
4
4
Rewrite the expression in terms of the first power of the cosine.
12.
cos 4 x
13.
sin 2 x cos 2 x
Verify the identity.
15.
sin u  sin v
1
 tan u  v 
cos u  cos v
2
17.
4 cos x cos 2 x sin 3x  sin 2 x  sin 4 x  sin 6x
18.
19.
sin 4   4 sin  cos  1  2 sin 2  
cos 3x  cos x
  tan 2 x
sin 3x  sin x
20.
Find the exact zeros of the function in the interval
21.
3 csc x  2  0
1  cos 10 y  2 cos 2 5 y
16.
cos 3
 1  4 sin 2 
cos 
0,2 .
22.
sin 2 x  3cos 2 x
23.
2sin 2 x  3sin x  1  0
24.
2sin 3x   3
25.
sin 6x  sin 2 x  0
26.
cos x  cos 3x  0
27.
cos x  cos 3x  0
28.
sin 4 x  sin x  0
TRIGONOMETRY
PRACTICE TEST:
CHAPTER 5
NAME: __________________________
TRIGONOMETRIC IDENTITIES
In order to receive full credit, you must show your work.
Use the trigonometric identities to find the remaining trigonometric functions of
1.
cos  
3
3
   2
, where
5
2
2.
x or  .
2
2 5
sin  x    , tan x  
3
5
Each expression simplifies to a constant, a single trigonometric function, a multiple angle
trigonometric function, or a power of a trigonometric function. Use the trigonometric identities to
simplify each expression.
3.
sin x  cos x cot x
4.
5.
sec 2   tan 2 
6.
tan 2  1  cot 2  
2 sin  cos 
1  2 sin 2 
Find the exact value of sin x, cos x, and tan x for the given angle using
a)
sum or difference identities
b)
half-angle identities
7.
x

8.
12
105
Use the identities to express each expression to a single function of  .
9.
cos90   
10.
For each of the following, find
11.
cos x  
 5

tan 
 
 4

sin x  y , cosx  y , tan x  y , and the quadrant of x  y  .
15
4
and sin y  , where x is in Quadrant II and y is in Quadrant I.
17
5
Use the identities to find values of the six trigonometric functions for each of the following.
12.

, given cos 2
Find the exact values of
14.
cos A 

1
, where 270    360 13.
8
sin
5
2 x , given tan x  , sin x  0
3
A
A
A
, cos and tan using the half-angle identities.
2
2
2
3

, where 0  A 
5
2
Write the trigonometric expression as a product.
15.
cos 60  cos 30
16.
sin 5  sin 3
Write the trigonometric expression as a sum or difference.
17.
x
x
cos cos
2
4
18.
sin 120 cos 30
Rewrite the expression in terms of the first power of the cosine.
19.
sin 4 x cos 2 x
Find the exact zeros of the function in the interval
20.
2sin 2 x  2  cos x
22.
cos x  cos3x  0
0, 2 
21.
tan 3x  tan x 1  0
Verify the identity.
23.
sin 2 
 sec   cos 
cos 
24.
sin 3   sin  cos 2   sin 
25.
sec x  1
x
 tan
tan x
2
26.
sin    
 tan   tan 
cos  cos 
27.
cos 3  cos 3   3 sin 2  cos 
28.
1
x 1
x
cot  tan  cot x
2
2 2
2