1. No category

Trigonometry
Sample Final Exam 2012 -2013
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Provide an appropriate response.
1) Find the complement of an angle whose measure is 75°.
A) 165°
B) 15°
C) 105°
1)
D) 75°
Find the measure of each angle in the problem.
2)
A) 100° and 80°
2)
B) 95° and 85°
C) 50° and 40°
D) 200° and 160°
Convert the angle to decimal degrees and round to the nearest hundredth of a degree.
3) 186°2′51′′
A) 186.06°
B) 186.05°
C) 186.01°
D) 186.11°
3)
Convert the angle to degrees, minutes, and seconds.
4) 144.62°
A) 144°35′ 62′′
B) 144°38′12′′
4)
C) 144°37′ 12′′
D) 144°37′62′′
Draw the given angle in standard position. Draw an arrow representing the correct amount of rotation. Find the
measure of two other angles, one positive and one negative, coterminal with the given angle.
5) 170°
5)
A)
350° and -10°
B)
530° and -190°
C) 530° and -190°
D) 350° and -10°
Solve the problem. Round answers to the nearest tenth if necessary.
6) A triangle drawn on a map has sides of lengths 8 cm, 11 cm, and 15 cm. The shortest of the
corresponding real-life distances is 140 km. Find the longest of the real-life distances.
A) 262.5 km
B) 182.5 km
C) 272.5 km
D) 192.5 km
Use the properties of angle measures to find the measure of each marked angle.
7) Find the measure of the marked angles.
a = (x + 78)°
b = (x + 26)°
A) 19°, 38°, 33°
B) 19°, 38°, 123°
C) 38°, 97°, 45°
1
6)
7)
D) 71°, 71°, 38°
Classify the triangle as acute, right, or obtuse and classify it as equilateral, isosceles, or scalene.
8)
A) Acute, equilateral
C) Right, scalene
B) Obtuse, equilateral
D) Obtuse, scalene
The triangles are similar. Find the missing side, angle or value of the variable.
9)
a = 13
b = 12
c=5
d = 26
e = 24
A) x = 7
8)
B) x = 10
C) x = 5
9)
D) x = 15
If r is a positive number and the point (x, y) is in the indicated quadrant, decide whether the given ratio is positive or
negative.
r
10) III,
10)
x
A) Positive
B) Negative
Suppose that θ is in standard position and the given point is on the terminal side of θ. Give the exact value of the
indicated trig function for θ.
11) (4, 5); Find tan θ.
11)
5
4
5
2
A)
B)
C)
D)
6
5
4
3
An equation of the terminal side of an angle θ in standard position is given along with the quadrant of the angle θ.
Find the indicated trigonometric function value of θ. Do not use a calculator.
12) -2x + y = 0, x ≥ 0; Find sin θ.
12)
5
2 5
1
A) 2
B)
C)
D)
5
5
2
Use the appropriate identity to find the indicated function value. Rationalize the denominator, if applicable. If the
given value is a decimal, round your answer to three decimal places.
5
13) tan θ, given that cot θ = 13)
6
A) -
5
6
B)
11
6
C)
2
6
5
D) -
6
5
Sketch an angle θ in standard position such that θ has the smallest positive measure and the given point is on the
terminal side of θ.
14) (-5, -3)
14)
y
x
A)
B)
C)
D)
Evaluate the expression.
15) sin(-180°)
A) 0
15)
B) -1
16) cot 90° + 2 cos 180° + 6 sec2 360°
A) -2
B) 8
C) Undefined
D) 1
C) 7
D) 4
16)
Identify the quadrant for the angle θ satisfying the following conditions.
17) cos θ < 0 and csc θ < 0
A) Quadrant I
B) Quadrant III
C) Quadrant IV
Decide whether the statement is possible or impossible for an angle θ.
18) sin θ = 4.14
A) Possible
B) Impossible
3
17)
D) Quadrant II
18)
Use the fundamental identities to find the value of the trigonometric function.
3
19) Find tan θ, given that sin θ = and θ is in quadrant II.
4
A) -
3 7
7
B) -
20) Find cot θ, given that cos θ =
A) -
21 2
4
3
2
C) -
7
9
19)
D)
5
4
21
and θ is in quadrant IV.
29
B)
29
21
C) -
20)
20
21
D) -
21
20
Write the function in terms of its cofunction. Assume that any angle in which an unknown appears is an acute angle.
21) sin 7°
21)
A) cos 83°
B) sin 97°
C) cos 7°
D) csc 83°
Evaluate the function requested. Write your answer as a fraction in lowest terms.
22)
22)
41
9
40
Find cos B.
A) cos B =
40
41
B) cos B =
9
41
C) cos B =
9
40
D) cos B =
41
40
Suppose ABC is a right triangle with sides of lengths a, b, and c and right angle at C. Find the unknown side length
using the Pythagorean theorem and then find the value of the indicated trigonometric function of the given angle.
Rationalize the denominator if applicable.
23) Find cos A when a = 14 and c = 9.
23)
67
67
14
14
A)
B)
C)
D)
9
14
9
9
Find a solution for the equation. Assume that all angles are acute angles.
24) sin A = cos 4A
A) 4°
B) 18°
C) 72°
4
24)
D) 86°
Solve the problem.
25) Find the exact value of x in the figure.
25)
14
x
A) 7 6
B) 7 3
C)
Find the reference angle for the given angle.
26) 211.4°
A) 121.4°
B) 58.6°
14 6
3
D)
14 3
3
26)
C) 31.4°
D) 148.6°
Give the exact value.
27) sec 150°
A)
2
27)
B) -
C) -
2
Find the exact trigonometric function value.
28) cos 960°
2
A) - 3
B)
2
2 3
3
D)
2 3
3
28)
C) -
1
2
D) -
3
2
Find all values of θ, if θ is in the interval [0, 360°) and has the given function value.
1
29) cos θ =
2
A) 60° and 120°
B) 210° and 330°
C) 60° and 300°
29)
D) 150° and 210°
Use a calculator to find the function value. Give your answer rounded to seven decimal places, if necessary.
30) sin 285°3′
30)
A) -0.9556996
B) -0.9656996
C) -0.9666996
D) -0.9756996
31) sec 15°44′
A) 2.0778484
31)
B) 2.0788484
C) 1.0389242
D) 1.0379242
Find a value of θ in [0°, 90°] that satisfies the statement. Leave answer in decimal degrees rounded to seven decimal
places, if necessary.
32) sin θ = 0.69771214
32)
A) 224.243736°
B) 45.7562643°
C) 135.756264°
D) 44.2437357°
5
33) csc θ = 1.3749283
A) 43.3385921°
33)
B) 53.9712054°
C) 46.6614079°
Solve the right triangle.
34) B = 42.1°, c = 4.2 mm, C = 90°
A) a = 3.1 mm, A = 47.9°, b = 2.8 mm
C) a = 2.3 mm, A = 47.9°, b = 3.5 mm
D) 36.0287946°
34)
B) a = 3.1 mm, A = 47.9°, b = 2.3 mm
D) a = 2.8 mm, A = 47.9°, b = 3.1 mm
Solve the problem.
35) On a sunny day, a flag pole and its shadow form the sides of a right triangle. If the hypotenuse is
52 meters long and the shadow is 48 meters, how tall is the flag pole?
A) 100 m
B) 32 m
C) 20 m
D) 71 m
35)
36) From a boat on the lake, the angle of elevation to the top of a cliff is 29°38'. If the base of the cliff is
655 feet from the boat, how high is the cliff (to the nearest foot)?
A) 383 ft
B) 386 ft
C) 376 ft
D) 373 ft
36)
37) From a balloon 947 feet high, the angle of depression to the ranger headquarters is 73°44'. How far
is the headquarters from a point on the ground directly below the balloon (to the nearest foot)?
A) 271 ft
B) 281 ft
C) 266 ft
D) 276 ft
37)
Convert the degree measure to radians. Leave answer as a multiple of π.
38) 288°
8π
9π
4π
A)
B)
C)
5
5
5
38)
16π
D)
5
Convert the radian measure to degrees. Round to the nearest hundredth if necessary.
10π
39)
3
A) 600°
B) 601°
C) 599.5°
39)
D) 600.5°
Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π.
40) 28°
A) 0.3258
B) 0.9774
C) 0.2443
D) 0.4887
40)
Convert the radian measure to degrees. Give answer using decimal degrees to the nearest hundredth. Use 3.1416 for π.
41) 0.2521
41)
A) 13.74°
B) 14.94°
C) 14.44°
D) 15.44°
Find the exact value without using a calculator.
3π
42) sin
4
A) -
2
2
B) -
42)
1
2
C)
6
2
2
D)
1
2
43) cot
-5π
6
43)
3
3
A)
B)
C) -
3
3
3
D) - 3
Find the length of an arc intercepted by a central angle θ in a circle of radius r. Round your answer to 1 decimal place.
π
44) r = 38.81 ft; θ = radians
44)
3
A) 81.3 ft
B) 20.3 ft
C) 40.6 ft
D) 121.9 ft
Find the area of a sector of a circle having radius r and central angle θ. Express the answer to the nearest tenth.
π
45) r = 67.4 cm, θ = radians
45)
7
A) 1019.4 cm 2
B) 2038.8 cm2
Use a table or a calculator to evaluate the function.
46) tan 0.2229
A) 1.025
B) 0.2211
C) 324.5 cm 2
D) 15.1 cm 2
C) 0.2267
D) 0.9753
46)
Find the value of s in the interval [0, π/2] that makes the statement true.
47) sec s = 2.5905236
A) 0.39631581
B) 5.10870479
C) -0.42920367
47)
D) 1.17448051
Find the exact value of s in the given interval that has the given circular function value.
π
3
48)
, π ; cos s = 2
2
A) s =
49)
3π
4
B) s =
5π
6
C) s =
2π
3
48)
D) s =
π
6
3π
3
, 2π ; tan s = 2
3
A) s =
7π
4
49)
B) s =
π
6
C) s =
5π
3
D) s =
Give the amplitude or period as requested.
1
50) Amplitude of y = -2 cos x
3
A)
2π
3
50)
B) 2
C) 6π
D)
π
2
51) Period of y = sin 3x
A) 1
11π
6
51)
B) 2π
C)
Find the specified quantity.
52) Find the amplitude of y = 5 cos (3x - π).
A) 5
B) π
2π
3
D) 3
52)
C) 3
7
D) 15
53) Find the period of y = -2 sin 4x +
A) π
π
.
2
53)
B) 2
C)
π
2
D) 4
Match the function with its graph.
2) y = 3 cos x
54) 1) y = sin 3x
3) y = 3 sin x
4) y = cos 3x
54)
A)
B)
3
-2
y
3
2
2
1
1
-

2
x
-2
y
-
-1
-1
-2
-2
-3

2

2
x
-3
C)
D)
y
-2
-
y
3
3
2
2
1
1
-1

2
x
-2
-
-1
-2
-2
-3
-3
A) 1A, 2D, 3C, 4B
B) 1B, 2D, 3C, 4A
C) 1A, 2C, 3D, 4B
x
D) 1A, 2B, 3C, 4D
Find the specified quantity.
55) Find the vertical translation of y = -3 - 2 sin 4x +
A) up
1
4
π
.
4
B) up 4
55)
C) up
π
4
D) down 3
Solve the problem.
56) The voltage E in an electrical circuit is given by E = 4.3 cos 150πt, where t is time measured in
seconds. Find the frequency of the function (that is, find the number of cycles or periods
completed in one second).
1
1
A) 150
B)
C) 75
D)
150
75
8
56)
Match the function with its graph.
π
π
57) 1) y = sin x 2) y = cos x +
2
2
3) y = sin x +
π
2
4) y = cos x -
A)
π
2
B)
3
-2
57)
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
C)
y

2

2
x
D)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
A) 1A, 2D, 3C, 4B
B) 1A, 2B, 3C, 4D
y
C) 1C, 2A, 3B, 4D
x
D) 1B, 2D, 3C, 4A
Solve the problem.
58) The temperature in Fairbanks is approximated by
T(x) = 37 sin
58)
2π
(x - 101) + 25,
365
where T(x) is the temperature on day x, with x = 1 corresponding to Jan. 1 and x = 365
corresponding to Dec. 31. Estimate the temperature on day 10.
A) -12°
B) -2°
C) -25°
D) -37°
9
Find the phase shift of the function.
π
59) y = -3 + 2 sin 3x 6
59)
A)
π
units to the right
18
B)
π
units to the right
12
C)
π
units to the left
6
D)
π
units to the left
18
Graph the function over a one-period interval.
60) y = 3 + sin(2x-π)
60)
y
5
4
3
2
1

4
3
4

2
5
4

3
2
7 x
4
A)
B)
y
y
5
5
4
4
3
3
2
2
1
1

4

2
3
4

5
4
3
2
7 x
4
C)

4

2
3
4

5
4
3
2
7 x
4

4

2
3
4

5
4
3
2
7 x
4
D)
y
y
5
5
4
4
3
3
2
2
1
1

4

2
3
4

5
4
3
2
7 x
4
10
Graph the function.
3
π
61) y = - cos x 4
3
61)
y
3
3 x
-3
-3
A)
B)
3
y
3
3 x
-3
y
3 x
-3
-3
-3
C)
D)
3
y
3
3 x
-3
y
3 x
-3
-3
-3
Use the fundamental identities to find the value of the trigonometric function.
8
62) Find sin θ if sec θ = - and tan θ < 0.
5
A) -
5
8
B)
39
8
C) -
11
39
8
62)
D)
39
64
Match the function with its graph.
π
π
63) 1) y = - tan x 2) y = tan x +
2
2
3) y = - cot x -
π
2
4) y = cot x +
63)
π
2
A)
B)
6
y
6
3
-2
y
3
-

2
x
-2
-
-3
-3
-6
-6
C)

2

2
x
D)
6
y
6
3
-2
-
3

2
x
-2
-
-3
-3
-6
-6
A) 1C, 2B, 3D, 4A
y
B) 1A, 2B, 3C, 4D
C) 1A, 2D, 3B, 4C
Use the fundamental identities to find the value of the trigonometric function.
64) Find cos θ if tan θ = 3 and sin θ < 0.
10
1
A) B)
C) - 10
10
3
x
D) 1D, 2A, 3C, 4B
64)
D) -
10
3
Factor the trigonometric expression and simplify.
65) 1 - 2 sin2 x + sin 4 x
A) (1 + tan 2 x)
65)
B) sin 2 x
C) cos4 x
12
D) (1 - sin2 x)
Graph the function.
4
2
π
66) y = cot x +
5
3
2
66)
y
3
-

x
-3
A)
B)
3
y
-
3

x
y
-
-3

x
-3
C)
D)
3
-
y
3

x
y
-
-3

x
-3
Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression.
67) sin 2 x + sin 2 x cot2 x
67)
A) sin 2 x + 1
B) cot2 x + 1
C) cot2 x - 1
13
D) 1
68)
sin 2 x - 1
cos (-x)
68)
A) -cos x
B) sin x
D) -sin x
C) cos x
Match the function with its graph.
69) 1) y = sec x
2) y = csc x
3) y = -sec x
4) y = -csc x
69)
A)
B)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
C)
y

2

2
x
D)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
A) 1C, 2A, 3B, 4D
B) 1A, 2D, 3C, 4B
y
C) 1B, 2D, 3C, 4A
x
D) 1A, 2B, 3C, 4D
Perform the indicated operations and simplify the result.
sin θ
sin θ
70)
1 + sin θ 1 - sin θ
A) -2 tan2 θ
70)
B) sin θ tan θ
C) 1 + cot θ
D) sec θ csc θ
B) - sec2 θ
C) 0
D) 1
71) tan θ(cot θ - cos θ)
A) 1 - sin θ
71)
Use the fundamental identities to simplify the expression.
72) sin 2 θ + tan 2θ + cos2 θ
A) tan 2 θ
72)
B) sec2 θ
C) cos3 θ
14
D) sin θ
73)
tan θ
sec θ
73)
A) tan 2 θ
B) sec2 θ
D) cos3 θ
C) sin θ
Graph the function.
1
6
π
74) y = csc x 2
5
6
3
74)
y
2
1
-2
-

2
x
-1
-2
-3
A)
B)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
C)
y

2

2
x
D)
3
-2
y
3
2
2
1
1
-

2
x
-2
-
-1
-1
-2
-2
-3
-3
y
Simplify the expression.
sin x cos x
75)
tan x
A) sin 2 x
x
75)
C) cos2 x
B) cos x
15
D) sin x
Graph the expression on each side of the equals symbol to determine whether the equation might be an identity.
76) (csc θ + cot θ)(1 - cos θ) = sin θ
76)
A) Not an Identity
B) Identity
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify that each equation is an identity.
1- sec θ
tan θ
77)
+
= -2 csc θ
tan θ
1 - sec θ
78)
77)
1 + cos t 1 - cos t
= 4 cot t csc t
1 - cos t 1 + cos t
78)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use Identities to find the exact value.
79) cos 165°
- 6- 2
A)
B)
4
79)
64
2
24
C)
Write in terms of the cofunction of a complementary angle.
80) sin 24°
A) cos 156°
B) csc 156°
6
D)
6+
4
2
80)
C) cos 66°
D) csc 66°
Use the cofunction identities to find an angle θ that makes the statement true.
81) sec (6θ + 17°) = csc (2θ - 7°)
83°
17°
A) θ = 40°
B) θ =
C) θ =
7
7
81)
D) θ = 10°
Use identities to write each expression as a function of θ.
π
82) cos θ +
2
A) sin θ
82)
B) -cos θ
C) -sin θ
D) cos θ
Find the exact value of the expression using the provided information.
1
1
83) Find cos(s + t) given that cos s = , with s in quadrant I, and sin t = - , with t in quadrant IV.
3
2
A)
2 6+1
6
B)
3+2 2
6
C)
Find the exact value by using a sum or difference identity.
84) sin 15°
6+ 2
- 6- 2
A)
B)
4
4
85) tan 75°
A) -
2 6-1
6
D)
3-2 2
6
D)
64
83)
84)
C)
-
6+
4
2
2
85)
3+2
B) -
3-2
C)
16
3+2
D)
3-2
Use trigonometric identities to find the exact value.
86) sin 25° cos 35° + cos 25° sin 35°
3
3
A)
B)
3
2
1
C)
2
5
D)
12
Use Identities to find the exact value.
87) cos 36° cos 24° - sin 36° sin 24°
2
A)
B) 1
2
3
C)
2
1
D)
2
86)
87)
Use trigonometric identities to find the exact value.
tan 10° + tan 20°
88)
1 - tan 10° tan 20°
A) 2
B)
88)
3
C)
1
2
D)
3
3
Find the exact value of the expression using the provided information.
1
1
89) Find sin(s - t) given that cos s = , with s in quadrant I, and sin t = - , with t in quadrant IV.
3
2
3-2 2
6
A)
B)
2 6-1
6
C)
2 6+1
6
D)
89)
3+2 2
6
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify that the equation is an identity.
90) sin(x + y) - sin(x - y) = 2 cos x sin y
90)
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use an identity to write the expression as a single trigonometric function or as a single number.
91) 2 cos2 22.5° - 1
3
3
A)
2
2
B)
C)
3
D)
2
4
Use identities to find the indicated value for each angle measure.
21
92) sin θ =
, cos θ > 0
Find cos(2θ).
29
A)
41
841
93) cos θ =
A) -
12
, sin θ < 0
13
120
169
B) -
41
841
C)
92)
840
841
D) -
43
841
Find sin(2θ).
B)
91)
93)
119
169
C)
17
120
169
D) -
119
169
Give the degree measure of θ.
94) θ = arcsin (-1)
A) 0°
95) θ = arctan -
94)
B) 180°
C) 45°
D) -90°
3
3
A) 210°
96) θ = cot-1 3
A) 150°
95)
C) -150°
B) 30°
D) -30°
96)
B) 60°
Use a calculator to give the real number value.
97) y = arcsec (2.8842912)
A) 69.714051
B) 1.2167397
C) 30°
D) 330°
C) 1.5139822
D) .35405662
97)
Evaluate the expression.
98) sin (arctan 2)
A) 5 2
98)
B) 2 5
C)
2 5
5
D)
5 2
2
Solve the equation for the interval [0, 2π).
99) sin 2 x - cos2 x = 0
99)
A)
π π
,
4 3
π π
B)
,
4 6
C)
π 3π 5π 7π
,
,
,
4 4 4 4
D)
π
4
Solve the equation in the interval [0°, 360°).
100) 4 sin 2 θ = 3
100)
B) ∅
D) {60°, 120°, 240°, 300°}
A) {60°, 120°}
C) {240°, 300°}
101) sin 2θ = -sin θ
A) {0°, 120°, 180°, 240°}
C) {0°, 180°}
101)
B) {0°, 60°, 120°, 180°, 240°, 300°}
D) {60°, 120°, 240°, 300°}
Solve the problem.
102) Points A and B are on opposite sides of a lake. Point C is 80.6 meters from A. The measure of angle
BAC is 76°40', and the measure of angle ACB is determined to be 38°30'. Find the distance
between points A and B (to the nearest meter).
A) 53 m
B) 55 m
C) 28 m
D) 26 m
Determine the number of triangles ABC possible with the given parts.
103) a = 39, b = 79, A = 70°
A) 1
B) 2
C) 0
18
102)
103)
D) 3
Solve the equation for solutions in the interval [0, 2π).
3
104) sin 4x =
2
104)
A)
π π 2π 7π 7π 13π 5π 19π
, ,
,
,
,
,
,
12 6 3 12 6 12 3
12
B) {0}
C)
π 5π
,
4 4
D) 0,
π
,π
4
Solve the triangle.
105)
105)
26 m
A) C = 103°, a = 11.7 m, b = 20.7 m
C) C = 103°, a = 20.7 m, b = 11.7 m
B) C = 97°, a = 11.5 m, b = 20.4 m
D) C = 103°, a = 57.8 m, b = 32.6 m
106)
106)
99.8 ft
A) B = 37.2°, b = 310.7 ft, c = 384.2 ft
C) B = 37.2°, b = 32 ft, c = 26 ft
B) B = 36.8°, b = 307.8 ft, c = 384.2 ft
D) B = 37.2°, b = 384.2 ft, c = 310.7 ft
Find the area of triangle ABC with the given parts. Round to the nearest whole number.
107) A = 41°40'
b = 13.1 m
c = 5.8 m
A) 50.6 m2
B) 52.6 m2
C) 25.3 m2
D) 12.7 m2
Find the missing parts of the triangle.
108) B = 42°30'
b = 8.67
a = 21.01
A) A = 38°15', C = 99°15', c = 26.68
C) no such triangle
107)
108)
B) A = 41°15', C = 98°15', c = 31.18
D) A = 40°15', C = 97°15', c = 29.68
Find the area of triangle ABC with the given parts. Round to the nearest whole number.
109) a = 17.4 cm
b = 15.0 cm
c = 13.4 cm
A) 106 cm 2
B) 100 cm2
C) 97 cm 2
D) 103 cm2
19
109)
Find the missing parts of the triangle.
110) A = 26°
a = 35 km
b = 47 km
A) B1 = 118°, C1 = 36°, c1 = 71 km
110)
B) B = 36°, C = 118°, c = 71 km
B2 = 10°, C2 = 144°, c2 = 14 km
C) B1 = 36°, C1 = 118°, c1 = 71 km
B2 = 144°, C2 = 10°, c2 = 14 km
D) no such triangle
111) C = 116.7°
a = 7.50 km
b = 9.80 km
A) c = 20.6 km, A = 24.9°, B = 38.4°
C) No triangle satisfies the given conditions.
111)
B) c = 14.8 km, A = 26.9°, B = 36.4°
D) c = 17.7 km, A = 28.9°, B = 34.4°
Find the missing parts of the triangle. (Find angles to the nearest hundredth of a degree.)
112) a = 6.1 in.
b = 13.6 in.
c = 16.0 in.
A) A = 23.92°, B = 54.34°, C = 101.74°
B) A = 19.92°, B = 56.34°, C = 103.74°
C) A = 21.92°, B = 56.34°, C = 101.74°
D) No triangle satisfies the given conditions.
Solve the problem.
113) Two points A and B are on opposite sides of a building. A surveyor chooses a third point C 66 yd
from B and 91 yd from A, with angle ACB measuring 59.4°. How far apart are A and B (to the
nearest yard)?
A) 81 yd
B) 108 yd
C) 99 yd
D) 90 yd
Write the number as the product of a real number and i.
114) -25
A) -i 25
B) 5i
D) 5 i
C) -i
D) i
115)
B) -1
Perform the indicated operations. Simplify the answer.
-72
116)
8
A) -3
116)
C) -3i
B) 3
D) 3i
Write the complex number in rectangular form.
π
π
117) 8 cos + i sin
6
6
A) 4 3 + 4i
113)
114)
C) -5i
Simplify the power of i.
115) i49
A) 1
112)
117)
B) 4 + 4 3i
C)
20
1
3
i
+
4
4
D)
3 1
+ i
4
4
118) 9 cis 240°
-9 3 9i
A)
2
2
118)
9 9i 3
B) - +
2
2
9 9i 3
C) 2
2
9 9i 3
D) - 2
2
Write the complex number in trigonometric form r(cos θ + i sin θ), with θ in the interval [0°, 360°).
119) 4 3 - 4i
A) 8(cos 30° + i sin 30°)
B) 8(cos 300° + i sin 300°)
C) 8(cos 330° + i sin 330°)
D) 8(cos 60° + i sin 60°)
119)
Find the product. Write the product in rectangular form, using exact values.
120) [8 cis 300°] [6 cis 330°]
A) 12 3 + 12i
B) 24
C) -48i
120)
D) -12 + 12 3i
Find the following quotient, and write the quotient in rectangular form, using exact values.
16(cos 240° + i sin 240°)
121)
4(cos 30° + i sin 30°)
A) - 2 + - 2 i 3
B) - 2
3-2i
C) - 2 3 - 2 i 3
D) - 2 - 2 i
C) -64
D) 64 - 64 3i
Find the given power. Write answer in rectangular form.
122) (- 3 + i)6
123) 2 cis 15° 4
A) 8 + 8 i
122)
B) -64 3 + 64i
A) 64i
121)
123)
3
B) 16i
C) 8
21
3+8i
D) 8 + 8 i
Graph the complex number.
124) -2 + i
124)
y
6
4
2
-6
-4
-2
2
6 x
4
-2
-4
-6
A)
B)
6
-6
-4
y
6
4
4
2
2
-2
2
4
6 x
-6
-4
-2
-2
-2
-4
-4
-6
-6
C)
y
2
4
6 x
2
4
6 x
D)
6
-6
-4
y
6
4
4
2
2
-2
2
4
6 x
-6
-4
-2
-2
-2
-4
-4
-6
-6
y
Find all cube roots of the complex number. Leave answers in trigonometric form.
125) 8(cos 318° + i sin 318°)
A) 2 cis 196°, 2 cis 256°, 2 cis 136°
B) 2 cis 106°, 2 cis 166°, 2 cis 46°
C) 2 cis 106°, 2 cis 226°, 2 cis 226°
D) 2 cis 106°, 2 cis 226°, 2 cis 346°
22
125)
Answer Key
Testname: SAMPLE FINAL EXAM 2011-2012
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)
48)
49)
50)
B
A
B
C
B
A
C
D
B
B
C
C
D
A
A
D
B
B
A
D
A
A
A
B
C
C
C
C
C
B
C
D
C
A
C
D
D
A
A
D
C
C
B
C
A
C
D
B
D
B
23
Answer Key
Testname: SAMPLE FINAL EXAM 2011-2012
51)
52)
53)
54)
55)
56)
57)
58)
59)
60)
61)
62)
63)
64)
65)
66)
67)
68)
69)
70)
71)
72)
73)
74)
75)
76)
C
A
C
B
D
C
C
A
A
A
B
B
D
A
C
B
D
A
A
A
A
B
C
D
C
B
77)
1- sec θ
tan θ
(1 - sec θ) 2 + tan 2θ 1 - 2 sec θ + sec2 θ + tan 2 θ 2 sec 2θ - 2 sec θ
2sec θ( sec θ - 1)
+
=
=
=
=
=tan θ
1 - sec θ
tan θ(1 - sec θ)
tan θ(1 - sec θ)
tan θ(1 - sec θ)
tan θ(1 - sec θ)
2sec θ
2
cos θ
2
=∙
== - 2 csc θ
tan θ
cos θ sin θ
sin θ
78)
1 + cos t 1 - cos t
(1 + cos t)2 - (1 - cos t)2
1 + 2 cos t + cos2 t - (1 - 2 cos t + cos2 t)
4 cos t
4 cos t
1
=
=
=
=
∙
1 - cos t 1 + cos t
2
2
2
sin
t
sin
t
1 - cos t
1 - cos t
sin t
79)
80)
81)
82)
83)
84)
85)
86)
87)
88)
89)
90)
91)
92)
93)
94)
= 4 cot t csc t
A
C
D
C
B
D
C
B
D
D
C
sin (x + y) - sin (x - y) = sin x cos y + cos x sin y - sin x cos y + cos x sin y = 2 cos x sin y.
B
B
A
D
24
Answer Key
Testname: SAMPLE FINAL EXAM 2011-2012
95)
96)
97)
98)
99)
100)
101)
102)
103)
104)
105)
106)
107)
108)
109)
110)
111)
112)
113)
114)
115)
116)
117)
118)
119)
120)
121)
122)
123)
124)
125)
D
C
B
C
C
D
A
B
C
A
A
A
C
C
C
C
B
C
A
B
D
D
A
D
C
C
B
C
A
A
D
25