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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 if cos = and is in quadrant IV.
3
5
3
A) -
2) Find cos
1
A)
3
B)
if tan
5
4
C) -
< 0.
10
B) 3
3
2
1)
D)
3 7
7
= 3 and sin
2)
C) -
10
10
D) - 10
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
3) (1 + cot )(1 - cot ) - csc2
3)
A) 2 cot2
4) sin2 x(cot2 x + 1)
A) tan2 x
5)
C) -2 cot2
B) 0
D) 2
C) cos2 x + 1
B) -1
4)
D) 1
sin2 x - 1
cos (-x)
5)
A) -sin x
B) cos x
C) sin x
D) -cos x
Perform the indicated operations and simplify the result so there are no quotients.
sin
sin
6)
1 + sin
1 - sin
A) sin
tan
- (1 + tan )2
7) 2 tan
A) 0
B) -2 tan 2
C) 1 + cot
D) sec
B) - sec2
C) 1
D) 1 - sin
Use Identities to find the exact value.
8) cos 165°
- 6- 2
A)
B)
4
9) cos A)
6)
csc
7)
8)
24
6
C)
6+
4
2
D)
64
2
7
12
64
9)
2
B)
24
6
C)
1
2-
6
D)
6+
2
Find the exact value of the expression using the provided information.
1
1
10) Find cos(s + t) given that sin s = - , with s in quadrant IV, and sin t = , with t in quadrant II.
2
4
A)
1+3 5
8
B)
4 3 - 15
11
C)
1-3 5
8
Use the cofunction identities to find an angle that makes the statement true.
11) tan (2 - 140°) = cot ( + 5°)
A) = 6°
B) = 75°
C) = 16°
D) -
4 3 + 15
11
D)
= 10°
11)
Use a sum or difference identity to find the exact value.
11
12) sin
12
A)
-
6+
4
2
13) tan 345°
A) 3 - 2
6+
4
B)
12)
2
C)
64
-
2
D)
64
D)
3+2
2
13)
B) -
3-2
3+2
C) -
Find the exact value of the expression using the provided information.
12
8
, with s in quadrant II, and sin t =
, with t in quadrant II.
14) Find sin(s + t) given that cos s = 13
17
A) -
21
221
B)
171
221
C) -
171
221
D)
24
25
15)
16)
B) 2 sin 4x
C) cos 8x
D) cos 4x
Use identities to find the indicated value for each angle measure.
4
Find cos(2 ).
< <2
17) sin = - ,
5
2
A) -
14)
21
221
Use an identity to write the expression as a single trigonometric function or as a single number.
15) sin 8x cos 8x
1
A) cos 8x
B) 2 sin 4x
C) sin 16x
D) cos 4x
2
16) 1 - 2 sin2 2x
1
A) sin 16x
2
10)
B) -
7
25
C)
17)
7
25
Write the product as a sum or difference of trigonometric functions.
18) 2 cos 6x cos 2x
1
A) (cos 8x + cos 4x)
B) cos 8x + cos 4x
2
C) sin 8x + sin 4x
D) cos 4x - cos 8x
2
D)
24
25
18)
Find the exact value by using a half-angle identity.
19) cos 165°
1
1
2- 3
2+ 3
A)
B)
2
2
20) sin
1
C) 2
2-
3
1
D) 2
19)
2+
3
5
12
20)
A) -
1
2
2+
3
B)
1
2
2+
3
C)
1
2
2-
3
1
2
D) -
2-
Find the exact value of the real number y.
3
21) y = sin-1
2
A)
B)
3
21)
C)
4
D)
3
4
Give the exact value of the expression.
21
22) cot sin-1
35
A)
21
35
B)
23) cos 2 arcsin
A)
22)
35
21
C)
B)
1
8
C)
C)
2
,
3 3
6
,
28
21
7
8
D)
5
8
24)
Solve the equation for exact solutions over the interval [0, 2 ).
25) 2 sin2 x = sin x
C)
D)
23)
Write the following as an algebraic expression in u, u > 0.
24) cos(arctan u)
u2 - 1
A)
B) u u2 + 1
u2 - 1
A)
21
28
1
4
3
8
3
B)
5
6
u u2 + 1
u2 + 1
D)
u2 + 1
u2 + 1
3
2
,
, ,
2 2 3 3
D) 0, ,
6
,
5
6
Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary.
26) 4 sin2 = 3
A) {240°, 300°}
C)
B) {60°, 120°}
D) {60°, 120°, 240°, 300°}
3
25)
26)
Solve the equation (x in radians and in degrees) for all exact solutions where appropriate. Round approximate
answers in radians to four decimal places and approximate answers in degrees to the nearest tenth.
27) cos2 x + 2 cos x = -1
27)
A) {n }
B) {2n }
C)
2
+ n , 2n
D) { + 2n }
28) 4 sin2 x - 1 = 0
A)
C)
28)
5
+n ,
+n
6
6
B)
5
+n
6
D)
+n ,
3
Solve the equation for solutions in the interval [0, 2 ).
29) sin2 2x = 1
A)
3
5
7
,
,
,
4 4
4
4
B)
2 cos 2x = 1
3
5
7
,
,
,
A)
4 4
4
4
C)
3
8
,
+n ,
2
+ 2n
+n
29)
9
,
8 8
D) 0,
C)
30)
6
2
4
, ,
3
3
30)
B)
9
7
15
,
,
8
8
8
D) 0,
2
4
, ,
3
3
Solve the equation for exact solutions.
15
31) cos-1 x = sin-1
17
A) {0}
32) arctan x = arccot
A)
B)
8
17
C)
D)
8
15
9
13
13
9
33) arccos x + arccos 2x = arccos
A) 2
31)
32)
B)
9
13
C)
D) {0}
1
2
33)
B) -5 + 3 2
C) -
4
3 3
,
4 4
D)
1
2
Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate.
34)
34)
28 m
A) C = 103°, a = 62.2 m, b = 35.1 m
C) C = 103°, a = 22.3 m, b = 12.6 m
B) C = 103°, a = 12.6 m, b = 22.3 m
D) C = 97°, a = 12.4 m, b = 21.9 m
Find the area of triangle ABC with the given parts. Round to the nearest tenth when necessary.
35) A = 41°40'
b = 17.5 m
c = 11.1 m
A) 64.6 m2
B) 32.3 m2
C) 131.2 m2
D) 129.2 m2
Determine the number of triangles ABC possible with the given parts.
36) b = 49, c = 59, B = 108°
A) 2
B) 0
C) 1
37) a = 35, b = 52, A = 29°
A) 0
B) 2
C) 1
D) 3
D) 3
Find the missing parts of the triangle.
38)
35)
36)
37)
38)
4
8
If necessary, round angles to the nearest degree and give exact values of side lengths.
A) B = 60°, C = 90°, b = 4 3
B) no such triangle
C) B = 60°, C = 90°, b = 4
D) B = 90°, C = 60°, b = 4 3
5
Find the indicated angle or side. Give an exact answer.
39) Find the exact length of side a.
39)
5 3
A)
109
B) 7
C)
47
D) 9
40) Find the measure of angle A in degrees.
40)
2 13
A) 120°
B) 60°
C) 135°
D) 140°
Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as
an angle in [0,360°].
41) -12, 5
41)
A) 13; 112.6°
B) 15; 157.4°
C) 13; 157.4°
D) 13; 22.6°
42) -4, -3
A) 5; 233.1°
B) 5; 216.9°
C) 7; 216.9°
D) 5; 36.9°
42)
Vector v has the given magnitude and direction. Find the magnitude of the indicated component of v rounded to the
nearest tenth when necessary.
43) = 39.4°, v = 206; Find the vertical component of v.
43)
A) 28.4
B) 290
C) 130.8
D) 159.2
6
Write the vector in the form <a, b>. If necessary, round values to the nearest hundredth.
44)
A) -8, -13.86
B) -16, -27.72
Find the dot product for the pair of vectors.
45) 1, 18 , 2, 1
A) 20
B) 22
C) -0.5, -0.87
D) -13.86, -8
C) 38
D) 16
Find the angle between the pair of vectors to the nearest tenth of a degree.
46) -5, 6 , 2, -1
A) 166.4°
B) 156.4°
C) 78.2°
Determine whether the pair of vectors is orthogonal.
47) 2, -1 , 16, 32
A) Yes
44)
45)
46)
D) 68.2°
47)
B) No
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify that each equation is an identity.
cos
48) sec + tan =
1 - sin
48)
Verify that the equation is an identity.
cos(x - y) 1 + tan x tan y
=
49)
cos(x + y) 1 - tan x tan y
49)
Verify that each equation is an identity.
1
50) cos4 x = (3 + 4 cos(2x) + cos (4x))
8
50)
7
Answer Key
Testname: UNTITLED1
1)
2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
15)
16)
17)
18)
19)
20)
21)
22)
23)
24)
25)
26)
27)
28)
29)
30)
31)
32)
33)
34)
35)
36)
37)
38)
39)
40)
41)
42)
43)
44)
45)
46)
47)
A
C
C
D
D
B
B
A
B
C
B
D
A
C
C
D
B
B
D
B
C
D
C
D
D
D
D
A
A
C
B
A
D
B
A
B
B
A
B
A
C
B
C
A
A
B
A
48) sec
+ tan
=
1
cos
+
sin
cos
=
1 + sin
cos
=
1 + sin
cos
·
1 - sin
1 - sin
8
=
1 - sin2
cos2
cos
=
=
cos (1 - sin ) cos (1 - sin )
1 - sin
Answer Key
Testname: UNTITLED1
49)
cos (x - y) cos x cos y + sin x sin y 1/(cos x cos y) cos x cos y + sin x sin y
=
=
·
=
cos (x + y) cos x cos y - sin x sin y 1/(cos x cos y) cos x cos y - sin x sin y
1 + tan x tan y
.
1 - tan x tan y
50) cos4 x = cos2 x cos2 x =
1 + cos(2x) 2 1
1
1
1 + cos(4x)
= (1 + cos(2x))2 = (1+ 2 cos(2x) + cos2 (2x)) = 1 + 2 cos(2x) +
=
2
4
4
4
2
1
( 3 + 4 cos(2x) + cos(4x)).
8
9