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Math 1316
Final Review
Name___________________________________
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the measures of two angles, one positive and one negative, that are coterminal with the given angle.
1) 112°
A) 472°, -68°
B) 382°, -158°
C) 292°, -68°
D) 472°, -248°
Convert the angle to radians. Leave as a multiple of π.
2) 570°
19π
19π
A)
B)
12
6
C)
19π
5
D)
19π
3
Solve the problem.
3) A wheel with a 24-inch diameter is turning at the rate of 32 revolutions per minute. To the nearest inch per
minute, what is the linear velocity of a point on the rim?
A) 2413 in./min
B) 2420 in./min
C) 2459 in./min
D) 2466 in./min
Given that α is an angle in standard position whose terminal side contains the given point, provide the exact value of the
indicated function.
4) (9, 12); cos α
3
3
4
4
A)
B)
C)
D)
5
4
5
3
Find the exact value of the following expression without using a calculator.
5) csc 60°
2 3
3
A)
B)
C) 2
3
2
6) tan
D) 2
7π
4
A) 1
B) -
3
3
C) -
2
2
D) -1
Use a calculator to find the function value to four decimal places.
7) cos (-711°)
A) 0.1564
B) 0.7771
C) 0.9877
D) 0.9336
Find the quadrant that contains the terminal side of angle α.
8) sec α < 0 and csc α < 0
A) I
B) II
C) III
D) IV
Solve the problem.
9) Find the acute angle α (in degrees) that satisfies the equation cos α =
A) 330°
B) 315°
C) 30°
1
2
.
2
D) 45°
Use a calculator to find the acute angle α (to the nearest tenth of a degree) that satisfies the equation.
10) sin α = 0.35167149
A) 159.4°
B) 69.4°
C) 200.6°
D) 20.6°
Evaluate the function requested. Write your answer as a fraction in lowest terms.
11) Find sin α.
45
27
36
A)
4
3
B)
3
5
C)
Solve the right triangle with the given sides and angles.
12) a = 3.6, β = 22.6°
A) α = 67.4°, b = 3.6, c = 5.1
C) α = 67.4°, b = 3.6, c = 3.9
4
5
D)
5
4
B) α = 67.4°, b = 1.5, c = 3.9
D) α = 67.4°, b = 0.4, c = 3.6
Solve the problem.
13) From a boat on the river below a dam, the angle of elevation to the top of the dam is 21°39'. If the dam is 2958
feet above the level of the river, how far is the boat from the base of the dam (to the nearest foot)?
A) 7422 ft
B) 7452 ft
C) 7442 ft
D) 7432 ft
Find the reference angle for the given angle.
14) 117°
A) 73°
B) 27°
C) 63°
D) 37°
Find the exact value of the trigonometric function.
5π
15) cos 4
A)
1
2
B)
3
2
C) -
2
3
2
D) -
2
2
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Graph the function over a one-period interval.
π
16) y = -3 cos x +
2
3
y
2
1
-1
π
2
π
3π
2
2π
x
-2
-3
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Determine the equation of the function that is graphed.
17)
y
5
-2π
2π
x
-5
A) y = 4 sin x
B) y = cos x + 4
Find the amplitude, period, or phase shift as specified.
18) Find the amplitude of y = -2 cos (4x - π).
A) -4
B) -8
3
C) y = sin (x + 4)
D) y = 4 cos x
C) π
D) 2
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Graph the function.
19) y = csc x +
π
3
3
-2π
y
-π
π
2π
π
2π
x
-3
20) y =
4
1
π
tan x +
5
3
2
3
-2π
-π
y
x
-3
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the fundamental identities to find an equivalent expression involving only sines and cosines, and then simplify it.
21) cot θ sin θ - tan θ cos θ
sin θ + cos θ
1
1
A)
B)
C) cos θ - sin θ
D)
sin θ cos θ
sin θ cos θ
sin θ cos2 θ
Use the fundamental identities to find the value of the trigonometric function.
2
22) Find sin α if cos α = and α is in quadrant IV.
3
A)
3 7
7
B) -
3
2
C)
5
4
D) -
5
3
Use the fundamental identities to simplify the expression.
1
23)
+ sec θ cos θ
cot2θ
A) tan2 θ
B) sec2 θ
C) 1
4
D) csc2 θ
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Verify the identify.
1 + csc x
24)
= cos x + cot x
sec x
Find the exact value by using a sum or difference identity.
5π
25) cos
12
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the sum/difference identities to simplify the expression. Do not use a calculator.
7π
2π
7π
2π
26) cos
cos
sin
+ sin
18
9
18
9
A) cos
π
6
B) cos
5π
6
C) cos
2π
3
D) cos
π
3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find cos(A - B).
27) cos A = -
5
8
and sin B =
, with A and B in quadrant II.
13
17
Find the exact value by using a sum or difference identity.
11π
28) sin
12
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a sum or difference identity to find the exact value.
7π
π
tan
- tan
24
8
29)
7π
π
1 + tan
tan
24
8
A)
3
B)
1
2
C)
3
3
D)
3
2
Solve the problem.
30) If cos A = A)
24
13
12
8
and sin B =
, with A and B in quadrant II, then find sin(A + B).
13
17
B)
171
221
C)
5
21
221
D) -
171
221
Find the exact value by using a half-angle identity.
π
31) cos 8
A)
1
2
2+
2
B)
1
2
2-
2
C)
1
2
C)
240
289
1+
2
D)
1
2
1-
2
Solve the problem.
32) Find sin 2θ. tan θ =
A)
8
, θ lies in quadrant III.
15
161
289
B) -
240
289
D) -
161
289
Find the exact value of the expression without using a calculator or table.
3
33) cos-1
2
A)
7π
4
B)
11π
6
C)
π
6
D)
π
4
Find the approximate value of the expression with a calculator. Round your answer to three decimal places.
34) tan-1 (-0.6536)
A) 3.721
B) -0.579
C) 2.150
D) 2.563
Find the exact value of the composition.
3
35) cot sin-1
5
A)
4
3
B)
3
5
C)
3
4
D)
Find the acute angle θ, to the nearest hundredth of a degree, for the given function value.
36) sin θ = 0.4965
A) -29.77°
B) 119.77°
C) 26.40°
Find all real numbers that satisfy the equation.
37) 2 cos x + 1 = 0
π
3π
A) x = + 2nπ or x =
+ 2nπ
2
2
C) x =
2π
4π
+ nπ or x =
+ nπ
3
3
5
3
D) 29.77°
B) x =
2π
4π
+ 2nπ or x =
+ 2nπ
3
3
D) x =
π
3π
+ nπ or x =
+ nπ
2
2
Find all values of θ in [0°, 360°) that satisfy the equation.
1
38) sin θ = 2
A) {60°, 300°}
B) {210°, 330°}
C) {60°, 120°}
6
D) {150°, 210°}
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find all real numbers in the interval [0, 2π) that satisfy the equation.
39) csc5 x - 4 csc x = 0
Find all values of x in the interval [0°, 360°) that satisfy the equation. Round approximate answers to the nearest tenth of a
degree.
40) sin2 x - 8 sin x - 4 = 0
Solve the triangle with the given parts.
41)
92.4
Solve the triangle. If there is more than one triangle with the given parts, give both solutions.
42) β = 26.9°
b = 19.02
a = 21.02
Solve the problem.
43) Two tracking stations are on the equator 118 miles apart. A weather balloon is located on a bearing of N 39°E
from the western station and on a bearing of N 22°E from the eastern station. How far is the balloon from the
western station? Round to the nearest mile.
Solve the triangle with the given information.
44) β = 63.5°
a = 12.20
c = 7.80
45) a = 6.8
b = 13.0
c = 16.6
Solve the problem.
46) A ship travels 54 km on a bearing of 13°, and then travels on a bearing of 103° for 156 km. Find the distance of
the end of the trip from the starting point, to the nearest kilometer.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the area of the triangle using Heron's formula. Round to the nearest unit.
47) a = 62
b = 88
c = 83.6
A) 3194
B) 2474
C) 2476
7
D) 3174
Find the magnitudes of the horizontal and vertical components for the vector v with the given magnitude and given
direction angle. Round to an appropriate number of significant digits.
48) ∣v∣ = 28.3, θ = 81.7°
A) vx = 4.1, vy = 28
B) vx = -28, vy = -4.1
C) vx = -4.1, vy = -28
D) vx = 28, vy = 4.1
Find the magnitude and direction angle (to the nearest tenth) of the vector. Give the measure of the direction angle as an
angle in [0°, 360°).
49) -12, 5
A) 13; 112.6°
B) 13; 22.6°
C) 15; 157.4°
D) 13; 157.4°
Perform the indicated operation. Use the form <a, b> for vectors.
50) v = <2, 7>, u = <2, 5>; Find 8v + 9u.
A) <32, 108>
B) <72, 63>
51) u = <11, -5>, v = <15, -14>; Find u · v.
A) 95
B) 70
C) <34, 101>
D) <648, 504>
C) 165
D) 235
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve.
52) Two forces of 80 N and 120 N (newtons) act on an object. The angle between the forces is 23°. Find the
magnitude of the resultant and the angle that it makes with the smaller force.
Solve the problem.
53) A force of 600 lb is required to pull a boat up a ramp inclined at 25.0° with the horizontal. How much does the
boat weigh?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the complex number in trigonometric form, using degree measure for the argument.
54) 5 3 - 5i
A) 10(cos(-60°) + i sin(-60°))
B) 10(cos(-30°) + i sin(-30°))
C) 10(cos 30° + i sin 30°)
D) 10(cos 150° + i sin 150°)
Write the complex number in the form a + bi.
5
55) (cos 150° + i sin 150°)
2
A) -
5 3 5
+ i
4
2
B) -
5 3 5
+ i
4
4
C) -
3 1
+ i
4
4
Perform the indicated operation. Write the answer in the form a + bi.
56) 3(cos 30° + i sin 30°) · 2(cos 90° + i sin 90°)
A) 2 6 - 2i
B) -6 - 6 3i
C) 3 - 12 3i
D) -
D) -3 + 3 3i
Use De Moivre's theorem to simplify the expression. Write the answer in a + bi form.
57) (- 3 + i)6
A) 64 - 64i 3
B) -64 3 + 64i
C) -64
8
3 1
+ i
4
2
D) 64i
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Find all specified roots.
58) Fourth roots of -16
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Convert to rectangular coordinates.
π
59) 12, 4
A) (-6 2, -6 2)
B) (6 2, -6 2)
C) (-6 2, 6 2)
D) (6 2, 6 2)
Plot the point whose polar coordinates are given.
60) (-4, -5π/4)
5
-5
5
-5
5
-5
A)
5
5
-5
-5
B)
9
5
-5
5
5
-5
5
-5
-5
C)
5
-5
D)
For the point given in rectangular coordinates, find equivalent polar coordinates (r, θ) for r > 0 and 0° ≤ θ < 360°.
61) (5 3, 5)
A) (20, 30°)
B) (5, 45°)
C) (10, 60°)
D) (10, 30°)
Graph the polar equation.
62) r = 8 sin 4θ
10
5
-10
-5
5
10
r
-5
-10
A)
B)
-10
10
10
5
5
-5
5
10
r
-10
-5
5
-5
-5
-10
-10
10
10
r
C)
D)
-10
10
10
5
5
-5
5
10
r
-10
-5
5
-5
-5
-10
-10
For the given polar equation, write an equivalent rectangular equation.
63) r = 10 sin θ
A) x2 + y2 = 10x
B) x2 + y2 = 10y
C)
x2 + y2 = 10y
10
r
D) x2 + y2 = 10x
Eliminate the parameter of the pair of parametric equations.
64) x = t + 4, y = t2
A) y =
x+x+4
B) y = x2 + 16
C) y =
11
x-4
D) y = x2 - 8x + 16
Answer Key
Testname: MATH 1316 FRSU11
1) D
2) B
3) A
4) A
5) A
6) D
7) C
8) C
9) D
10) D
11) C
12) B
13) B
14) C
15) D
16)
20)
3
-2π
-π
y
π
-3
3
21) C
22) D
23) B
24) Identity
2( 3 - 1)
25)
4
y
2
26) A
171
27)
221
1
-1
π
2
3π
2
π
2π
x
28)
29) C
30) D
31) A
32) C
33) C
34) B
35) A
36) D
37) B
38) B
π 3π 5π 7π
39)
,
,
,
4 4 4
4
-2
-3
17) A
18) D
19)
3
-2π
-π
2( 3 - 1)
4
y
π
2π
x
40) {208.2°, 331.8°}
41) β = 37.2°, b = 287.6, c = 355.7
42) α = 30.0°, γ = 123.1°, c = 35.2;
α' = 150.0°, γ' = 3.1°, c' = 2.27
43) 374 mi
44) b = 11.2, α = 77.6°, γ = 38.9°
45) α = 22.6°, β = 47.4°, γ = 109.9°
46) 165 km
47) B
48) A
49) D
50) C
51) D
-3
12
2π
x
Answer Key
Testname: MATH 1316 FRSU11
52) 196 N, 14°
53) 1420 lb
54) B
55) B
56) D
57) C
58) 2 + 2i, 2 59) B
60) C
61) D
62) C
63) B
64) D
2i, -
2+
2i, -
2-
2i
13