Exam 3 Review Ch 5
Math 1316
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the fundamental identities to find the value of the trigonometric function.
2
1) Find sin s if cos s = and s is in quadrant IV.
3
5
3
A) -
B)
2) Find sec s if tan s =
A) -
=
A) 2
4) Find cot
1
and
2
D)
5
4
=
A) 4
C)
5
4
D)
3 7
7
is in quadrant IV.
17
and
4
B)
3)
3
B) -
if csc
3
2
2)
7
9
B) -
if cos
C) -
3
and s is in quadrant I.
4
3
2
3) Find tan
3 7
7
1)
5
C) -
D) -
3
3
is in quadrant I.
4)
17
C)
1
4
D)
17
17
Complete the sentence so the result is an identity. Let x be any real number.
+ tan2 x = sec2 x
5)
5)
A) sin2x
B) -1
C) 1
D) cos2 x
= sin2 x
A) tan2 x
B) cot2x
C) cos2 x
D) sec2x
6) 1 -
6)
Use the fundamental identities to find an equivalent expression involving only sines and cosines, and then simplify it.
7) sec
+ sin
A)
1 + sin cos
cos
1 + sin2 cos2
B)
cos2
C)
1 + cos2
cos2
D) sin
8) sec2 + sin2
1 + sin2 cos2
A)
cos2
B)
C) sin
D)
1
1 + cos2
cos2
1 + sin cos
cos
7)
8)
Express the given trigonometric function in terms of the indicated function.
9) sin
in terms of cos
A) ± 1 - cos2
9)
1
B)
cos
C) ± 1 + cos2
D) 1 - cos
Show that the equation is not an identity by listing the value(s) of the variable from among 0,
, , and - for which
4 2
4
the equation is false.
10) (sin
+ cos )2 = 1
A) -
10)
B)
4
4
and -
C)
4
2
D) 0 and
2
Perform the indicated operations and simplify the result.
1
11) sec sec
A) 1 + cot
12)
sin
cos
+
B) sin
11)
C) -2 tan2
tan
D) sec
cos
sin
A) sec
12)
csc
C) -2 tan2
B) 1 + cot
D) sin
Factor the trigonometric expression.
13) 1 - 2 sin2 x + sin4 x
B) (1 - sin2 x)2 = cos4 x
D) (1 + tan2 x)
C) (1 - sin2 x)
14) sec4x - 2 sec2 x tan2 x + tan4 x
A) sec2x (1 + tan2 x)
14)
B) sec2x + tan2 x
C) 2
D) 1
Use the fundamental identities to simplify the expression.
csc cot
15)
sec
A) cot2
16) cos
- cos
A) cos3
tan
13)
A) sin2x
17)
csc
15)
B) sec2
C) 1
B) sec2
C) tan2
D) csc2
sin2
16)
D) sin
tan
cot
17)
A) tan2
B) cos3
C) sin
D) sec2
C) 1
D) tan2
Simplify the expression.
18) sin
cos sec
A) csc2
csc
18)
B) sec2
2
19) cos x tan x
A) 1
19)
B) sin x
C) cos x
D) cot x
Identify the equation as either an identity or not.
20) cot2x = (csc x - 1)(csc x + 1)
20)
A) Not an identity
B) Identity
Decide whether the expression is or is not an identity.
21) 1 - cos2 x = 1 - cos x
21)
A) Not an identity
B) Identity
Find the exact value by using a sum or difference identity.
5
22) cos
12
2( 3 - 1)
A)
22)
2( 3 - 1)
4
B) -
C)
2( 3 - 1)
4
2( 3 - 1)
D) -
23) cos 285°
23)
2( 3 - 1)
A) -
2( 3 + 1)
B) -
2( 3 - 1)
4
C) -
D)
2( 3 - 1)
4
Write in terms of the cofunction of a complementary angle.
24) cos
24)
12
A) sec
5
12
B) sin
5
12
C) sin
11
12
D) sec
11
12
25) tan 57°
25)
A) cot 123°
B) cot 147°
Use the cofunction identities to find an angle
26) tan
A)
C) cot 33°
D) tan 33°
that makes the statement true.
= cot (30°+ 5 )
= 6°
26)
B)
= 75°
C)
= 16°
D)
= 10°
B)
= 10°
C)
= 16°
D)
= 6°
27) sin (3 - 17°) = cos ( + 43°)
A)
= 90°
27)
Use the identities for the cosine of a sum or a difference to write the expression as a single function of x.
28) cos (90° - x)
A) -sin x
28)
B) sin x
C) cos x
D) -cos x
29) cos (x - 270°)
A) sin x
29)
B) cos x
C) -cos x
3
D) -sin x
Find the exact value of the expression using the provided information.
1
1
30) Find cos(A + B) given that cos A = , with A in quadrant I, and sin B = - , with B in quadrant IV.
3
2
A)
3-2 2
6
B)
2 6+1
6
186 18
B)
D)
2 6-1
6
3
5
, with A in quadrant IV, and sin B = , with B in
3
6
31) Find cos(A + B) given that sin A = quadrant IV.
55 - 15
A)
18
3+2 2
6
C)
15
C)
55 + 15
18
C)
2( 3 + 1)
4
30)
31)
D)
186 +
18
15
D)
2( 3 - 1)
4
Find the exact value by using a sum or difference identity.
32) sin 15°
A)
32)
2( 3 - 1)
4
33) sin 255°
2( 3 - 1)
A)
4
B)
2( 3 + 1)
4
-
33)
B)
2( 3 - 1)
4
-
2( 3 + 1)
4
C)
D)
2( 3 + 1)
4
-
34) tan 105°
A) -2 -
34)
3
B) 2 +
2+ 3
C)
4
3
2- 3
D)
4
Use trigonometric identities to find the exact value.
35) sin 20° cos 40° + cos 20° sin 40°
3
1
A)
B)
2
3
35)
C)
36) sin 245° cos 5° - cos 245° sin 5°
3
3
A)
B) 2
2
3
2
D)
1
3
36)
49
C)
12
1
D) 2
Use a sum or difference identity to find the exact value.
5
37) sin 12
A)
38) sin
64
2
B)
-
64
37)
2
C)
-
6+
4
2
D)
6+
4
2
11
12
A)
38)
64
2
B)
3+ 2
4
C)
4
3-2
4
D)
6+
4
2
Using a sum or difference identity, write the following as an expression involving functions of x.
39) sin
-x
4
39)
2
2
cos x +
sin x
2
2
A)
B) sin x
C) -cos x
40) sin x -
D)
40)
2
2
2
cos x +
sin x
2
2
A)
2
2
cos x sin x
2
2
B)
C) sin x
2
2
cos x sin x
2
2
D) -cos x
Find the exact value of the expression using the provided information.
1
1
41) Find sin(A - B) given that cos A = , with A in quadrant I, and sin B = - , with B in quadrant IV.
3
2
A)
2 6+1
6
B)
3-2 2
6
C)
2 6-1
6
D)
41)
3+2 2
6
Decide whether the expression is or is not an identity.
42) tan (A +
2
) = -cot A
42)
A) Identity
B) Not an identity
Determine if the equation is an identity.
43) sin(x + y) - sin(x - y) = 2 cos x sin y
43)
A) Identity
B) Not an identity
44) sin(x + y) - sin(x - y) = -2 cos x sin y
44)
A) Not an identity
B) Identity
Find the exact value.
45) sin 105°
2
A)
2
45)
B)
24
6
C)
64
2
D)
2+
4
6
Find the exact value, given that sin A = -4/5 with A in quadrant IV, tan B = 7/24 with B in quadrant III, and cos C = - 5/13
with C in quadrant II.
46) sin 2A
24
A) 25
24
B)
25
7
C) 25
5
7
D)
25
46)
47) tan 2A
7
A)
24
48) cos 2C
119
A) 169
7
B) 24
24
C) 7
24
D)
7
120
B) 169
119
C)
169
120
D)
169
Use an identity to write the expression as a single trigonometric function or as a single number.
49) 2 cos2 22.5° - 1
2
2
A)
50)
B)
3
3
2
4
C)
2 tan 15°
3
3
48)
49)
3
D)
50)
1 - tan215°
A)
47)
B)
2
2
3
C)
D)
2
4
Find the exact value of the function.
51) sin 2(150°)
3
A) 2
52) sin 2
51)
B) 0
C) -
3
1
D)
2
5
3
A) -
52)
3
53) tan 2(150°)
1
A)
2
B) -
3
2
C)
1
2
D) 0
53)
B) 0
C) -
3
Decide whether the expression is or is not an identity.
1 - tan x 1 - sin 2x
=
54)
1 + tan x
cos 2x
3
D) 2
54)
A) Identity
B) Not an identity
Write the product as a sum or difference of trigonometric functions.
55) cos 33° sin 14°
1
A) (cos 47° - cos 19°)
2
C)
1
B) (sin 47° - sin 19°)
2
1
(cos 47° + cos 19°)
2
D)
6
1
(sin 47° + sin 19°)
2
55)
56) 2 sin 4x sin 20x
1
A) (cos 24x + cos 16x)
2
56)
B) sin 24x + sin 16x
C) cos 16x - cos 24x
D) cos 24x + cos 16x
Find the exact value by using a half-angle identity.
57) sin 22.5°
1
2+
A)
2
58) cos 22.5°
1
A) 2
2-
1
B) 2
2
2
1
B)
2
2-
2+
1
C)
2
2
2-
1
C) 2
2
2+
57)
1
D) 2
2
2
2+
58)
1
D)
2
2-
59) tan 75°
3
B) -2 -
Find the function value.
60) Find cos
, given that cos
2
15
, given that sin
6
4
62) Find cos
A)
2
8-2
4
61) Find cos
A)
=
1
and
4
B)
=
, given that sec
6
4
8+2
4
B)
3
D) -2 +
3
15
C)
60)
6
4
D)
10
4
terminates in quadrant I.
8-2
4
= 4 and
C) 2 +
terminates in quadrant I.
1
and
4
B)
2
3
15
C)
8+2
4
61)
15
D)
10
4
terminates in quadrant I.
8-2
4
15
C)
8+2
4
62)
15
D)
10
4
Use an identity to write the expression as a single trigonometric function or as a single number.
sin 70°
63)
1 + cos 70°
A) cot 35°
64)
2
59)
A) 2 -
A)
2
B) cos 35°
C) tan 35°
D) sin 35°
1 - cos 22°
2
A) sin 11°
63)
64)
B) cot 11°
C) tan 11°
7
D) cos 11°
Decide whether the expression is or is not an identity.
x
x sin x
65) sin cos =
2
2
2
65)
A) Not an identity
B) Identity
x 1 - cos x
66) tan2 =
2 1 + cos x
66)
A) Identity
B) Not an identity
8