Trigonometry a unit circle approach 9th edition sullivan test bank
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Ch. 2 Trigonometric Functions
2.1 Angles and Their Measure
1 Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Draw the angle.
1) 60°
A)
B)
C)
D)
2) 135°
A)
B)
C)
D)
Page 1
3)
2π
3
4) -
A)
B)
C)
D)
3π
4
A)
B)
C)
D)
Page 2
5) -150°
A)
B)
C)
D)
6) 330°
A)
B)
C)
D)
Page 3
7) -
8)
7π
6
A)
B)
C)
D)
5π
3
A)
B)
C)
D)
Page 4
9) -120°
A)
B)
C)
10)
D)
7π
4
A)
B)
C)
D)
Convert the angle to a decimal in degrees. Round the answer to two decimal places.
11) 11°41ʹ45ʹʹ
A) 11.71°
B) 11.76°
C) 11.66°
Page 5
D) 11.70°
12) 140°49ʹ59ʹʹ
A) 140.83°
B) 140.79°
C) 140.89°
D) 140.84°
13) 270°8ʹ40ʹʹ
A) 270.15°
B) 270.10°
C) 270.20°
D) 270.14°
14) 23°47ʹ37ʹʹ
A) 23.94°
B) 23.84°
C) 23.79°
D) 23.52°
15) 21°17ʹ34ʹʹ
A) 21.22°
B) 21.34°
C) 21.29°
D) 21.37°
Convert the angle to D° Mʹ Sʹʹ form. Round the answer to the nearest second.
16) 39.08°
A) 39°4ʹ48ʹʹ
B) 39°4ʹ54ʹʹ
C) 39°4ʹ8ʹʹ
D) 39°4ʹ36ʹʹ
17) 175.32°
A) 175°20ʹ12ʹʹ
B) 175°19ʹ32ʹʹ
C) 175°19ʹ12ʹʹ
D) 175°17ʹ32ʹʹ
18) 265.43°
A) 265°26ʹ47ʹʹ
B) 265°25ʹ48ʹʹ
C) 265°25ʹ43ʹʹ
D) 265°47ʹ43ʹʹ
2 Find the Length of an Arc of a Circle
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
If s denotes the length of the arc of a circle of radius r subtended by a central angle θ, find the missing quantity.
1) r = 4.05 centimeters, θ = 6 radians, s = ?
A) 25.3 cm
B) 23.3 cm
C) 24.3 cm
D) 26.3 cm
2) r = 14.0 inches, θ = 30°, s = ?
A) 7.6 in.
B) 7.5 in.
C) 7.4 in.
B) 24°
C)
D) 7.3 in.
1
3) r = feet, s = 8 feet, θ = ?
3
A)
8
radians
3
4) s = 9.5 meters, θ = 2.5 radians, r = ?
A) 3 m
B) 3.8 m
8
°
3
C) 0.26 m
Page 6
D) 24 radians
D) 1.9 m
Find the length s. Round the answer to three decimal places.
5)
s
π
4
12 m
A) 9.425 m
B) 18.85 m
C) 1.047 m
D) 15.279 m
C) 5.026 cm
D) 2.513 cm
C) 6.911 cm
D) 6.143 cm
C) 2.618 m
D) 2.88 m
6)
π
5
s
4 cm
A) 3.927 cm
B) 6.366 cm
7)
s
55°
8 cm
A) 8.447 cm
B) 7.679 cm
8)
s
30°
5 m
A) 2.356 m
B) 2.094 m
Solve the problem.
9) For a circle of radius 4 feet, find the arc length s subtended by a central angle of 30 °. Round to the nearest
hundredth.
A) 6.28 ft
B) 2.09 ft
C) 4.19 ft
D) 376.99 ft
Page 7
10) For a circle of radius 4 feet, find the arc length s subtended by a central angle of 60 °. Round to the nearest
hundredth.
A) 4.25 ft
B) 4.35 ft
C) 4.40 ft
D) 4.19 ft
11) A ship in the Pacific Ocean measures its position to be 31°16ʹ north latitude. Another ship is reported to be due
north of the first ship at 38°26ʹ north latitude. Approximately how far apart are the two ships? Round to the
nearest mile. Assume that the radius of the Earth is 3960 miles.
A) 28,369 mi
B) 484 mi
C) 28,380 mi
D) 495 mi
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12) Salt Lake City, Utah, is due north of Flagstaff, Arizona. Find the distance between Salt Lake City (40 °45ʹ north
latitude) and Flagstaff (35°16ʹ north latitude). Assume that the radius of the Earth is 3960 miles. Round to
nearest whole mile.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
13) The minute hand of a clock is 6 inches long. How far does the tip of the minute hand move in 10 minutes? If
necessary, round the answer to two decimal places.
A) 6.28 in.
B) 4.54 in.
C) 8.79 in.
D) 7.51 in.
14) A pendulum swings though an angle of 30° each second. If the pendulum is 45 inches long, how far does its tip
move each second? If necessary, round the answer to two decimal places.
A) 24.85 in.
B) 21.71 in.
C) 25.99 in.
D) 23.56 in.
3 Convert from Degrees to Radians and from Radians to Degrees
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Convert the angle in degrees to radians. Express the answer as multiple of π.
1) 90°
π
π
π
B)
C)
A)
8
3
2
D)
π
4
2) -36°
A) -
π
7
B) -
π
5
C) -
π
6
D) -
π
4
3) 75°
A)
6π
13
B)
4π
11
C)
5π
12
D)
12π
5
4) -75°
A) -
4π
11
B) -
5π
12
C) -
6π
13
D) -
12π
5
5) 87°
A)
29π
30
B)
29π
90
C)
29π
60
D)
29π
120
A)
π
60
B)
π
15
C)
π
18
D)
π
30
6) 6°
Page 8
Convert the angle in radians to degrees.
6π
7)
9
A) 122°
8) -
9)
B) 120°
C) 119°
D) 121°
A) -449°
B) -452°
C) -450°
D) -451°
A) 60π°
B) 60°
C) 1°
D) 3°
B) -36°
C) -1°
D) 1°
B) 315°
C) 630°
D) 103π°
A) 240°
B) 480°
C) 8°
D) 960π°
A) 15°
B) 60°
C) 1080°
D) 30°
B) 165°
C) 210°
D) 160°
5π
2
π
3
10) -
π
5
A) -36π°
11)
7π
4
A) 154°
12)
13)
14)
8
π
3
π
6
11π
12
A) 150°
Convert the angle in degrees to radians. Express the answer in decimal form, rounded to two decimal places.
15) 45°
A) 0.78
B) 0.79
C) 0.77
D) 0.76
16) -239°
A) -4.14
B) -4.15
C) -4.17
D) -4.16
Convert the angle in radians to degrees. Express the answer in decimal form, rounded to two decimal places.
17) 2
A) 0.03°
B) 116.07°
C) 0.2°
D) 114.59°
18) 8.96
A) 513.22°
19)
6
A) 140.35°
B) 0.23°
C) 0.16°
D) 513.37°
B) 0.04°
C) -0.04°
D) 141.68°
Page 9
4 Find the Area of a Sector of a Circle
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
If A denotes the area of the sector of a circle of radius r formed by the central angle θ, find the missing quantity. If
necessary, round the answer to two decimal places.
π
1) r = 13 inches, θ = radians, A = ?
6
A) 44.22 in2
B) 88.44 in2
C) 6.8 in2
D) 3.4 in2
C) 19,600 radians
D) 0.51 radians
B) 21.98 m
C) 87.92 m
D) 11.94 m
B) 19.23 in2
C) 38.47 in2
D) 5.5 in2
5) r = 7 feet, A = 92 square feet, θ = ?
A) 215.26°
B) 258,420.38°
C) 129,210.19°
D) 107.63°
6) θ = 60°, A = 98 square meters, r = ?
A) 7.16 m
B) 51.29 m
C) 13.68 m
D) 205.15 m
C) 1069 cm 2
D) 2138 cm 2
C) 21.98 ft2
D) 40.96 ft2
C) 18.85 ft 2
D) 37.699 ft 2
2) r = 14 feet, A = 100 square feet, θ = ?
A) 1.02 radians
B) 9800 radians
π
3) θ = radians, A = 56 square meters, r = ?
4
A) 4.69 m
4) r = 7 inches, θ = 45°, A = ?
A) 2.75 in2
π
7) r = 63.9 centimeters, θ = radians, A = ?
6
A) 340.3 cm 2
B) 16.7 cm 2
8) r = 11.9 feet, θ = 15.361°, A = ?
A) 18.98 ft2
B) 37.96 ft2
Find the area A. Round the answer to three decimal places.
9)
π
3
6 ft
A) 3.142 ft 2
B) 12 ft 2
Page 10
10)
π
6
12 yd
A) 3.142 yd2
B) 24 yd2
C) 37.699 yd2
D) 75.398 yd2
B) 47.997 m 2
C) 4.8 m 2
D) 95.993 m 2
B) 21.817 yd2
C) 2.182 yd2
D) 6.944 yd2
11)
55°
10 m
A) 15.278 m 2
12)
25°
10 yd
A) 43.633 yd2
Solve the problem.
13) A circle has a radius of 12 centimeters. Find the area of the sector of the circle formed by an angle of 75°. If
necessary, round the answer to two decimal places.
A) 30 cm2
B) 188.5 cm2
C) 7.85 cm2
D) 94.25 cm2
14) An irrigation sprinkler in a field of lettuce sprays water over a distance of 30 feet as it rotates through an angle
of 120°. What area of the field receives water? If necessary, round the answer to two decimal places.
A) 942.48 ft 2
B) 1884.96 ft 2
C) 300 ft 2
D) 31.42 ft 2
15) As part of an experiment to test different liquid fertilizers, a sprinkler has to be set to cover an area of 110
square yards in the shape of a sector of a circle of radius 60 yards. Through what angle should the sprinkler be
set to rotate? If necessary, round the answer to two decimal places.
A) 2.63°
B) 1.75°
C) 11°
D) 3.5°
16) The blade of a windshield wiper sweeps out an angle of 135 ° in one cycle. The base of the blade is 12 inches
from the pivot point and the tip is 32 inches from the pivot point. What area does the wiper cover in one cycle?
(Round to the nearest 0.1 square inch.)
A) 1105.3 in2
B) 1036.7 in2
C) 1041.8 in2
D) 948.3 in2
Page 11
5 Find the Linear Speed of an Object Traveling in Circular Motion
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) An object is traveling around a circle with a radius of 10 centimeters. If in 20 seconds a central angle of
radian is swept out, what is the linear speed of the object?
1
1
B) radians/sec
A) cm/sec
6
6
C) 6 radians/sec
D) 6 cm/sec
2) An object is traveling around a circle with a radius of 20 meters. If in 10 seconds a central angle of
swept out, what is the linear speed of the object?
1
1
B) m/sec
A) m/sec
8
5
C)
1
m/sec
4
1
3
D)
1
radian is
5
2
m/sec
5
3) An object is traveling around a circle with a radius of 10 meters. If in 15 seconds a central angle of 3 radians is
swept out, what is the linear speed of the object?
1
2
C) 2 m/sec
D) m/sec
A) 3 m/sec
B) m/sec
3
3
4) A weight hangs from a rope 20 feet long. It swings through an angle of 27 ° each second. How far does the
weight travel each second? Round to the nearest 0.1 foot.
A) 9.4 feet
B) 9.0 feet
C) 8.1 feet
D) 8.7 feet
π
5) A gear with a radius of 2 centimeters is turning at radians/sec. What is the linear speed at a point on the
3
outer edge of the gear?
π
A) cm/sec
6
B) 6π cm/sec
C)
2π
cm/sec
3
D)
3π
cm/sec
2
6) A wheel of radius 5.2 feet is moving forward at 10 feet per second. How fast is the wheel rotating?
A) 0.6 radians/sec
B) 3.2 radians/sec
C) 1.9 radians/sec
D) 0.52 radians/sec
7) A car is traveling at 48 mph. If its tires have a diameter of 26 inches, how fast are the carʹs tires turning? Express
the answer in revolutions per minute. If necessary, round to two decimal places.
A) 3899.08 rpm
B) 1241.11 rpm
C) 620.56 rpm
D) 633.56 rpm
8) A pick-up truck is fitted with new tires which have a diameter of 42 inches. How fast will the pick-up truck be
moving when the wheels are rotating at 430 revolutions per minute? Express the answer in miles per hour
rounded to the nearest whole number.
A) 61 mph
B) 9 mph
C) 27 mph
D) 54 mph
9) The Earth rotates about its pole once every 24 hours. The distance from the pole to a location on Earth 53° north
latitude is about 2383.2 miles. Therefore, a location on Earth at 53° north latitude is spinning on a circle of
radius 2383.2 miles. Compute the linear speed on the surface of the Earth at 53° north latitude.
A) 99 mph
B) 14,974 mph
C) 601 mph
D) 624 mph
10) To approximate the speed of a river, a circular paddle wheel with radius 0.68 feet is lowered into the water. If
the current causes the wheel to rotate at a speed of 8 revolutions per minute, what is the speed of the current? If
necessary, round to two decimal places.
A) 34.18 mph
B) 0.39 mph
C) 0.06 mph
D) 0.19 mph
Page 12
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
11) The four Galilean moons of Jupiter have orbital periods and mean distances from Jupiter given by the
following table.
Distance (km) Period (Earth hours)
4.214 × 105
42.460
5
6.709 × 10
85.243
Io
Europa
Ganymeade
Callisto
1.070 × 106
1.883 × 106
171.709
400.536
Find the linear speed of each moon. Which is the fastest (in terms of linear speed)?
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
12) In a computer simulation, a satellite orbits around Earth at a distance from the Earthʹs surface of 2.4 x 104 miles.
The orbit is circular, and one revolution around Earth takes 10.6 days. Assuming the radius of the Earth is 3960
miles, find the linear speed of the satellite. Express the answer in miles per hour to the nearest whole mile.
A) 14,600 mph
B) 691 mph
C) 110 mph
D) 593 mph
13) A carousel has a radius of 19 feet and takes 27 seconds to make one complete revolution. What is the linear
speed of the carousel at its outside edge? If necessary, round the answer to two decimal places.
A) 4.42 ft/sec
B) 0.7 ft/sec
C) 119.38 ft/sec
D) 8.93 ft/sec
2.2 Trigonometric Functions: Unit Circle Approach
1 Find the Exact Values of the Trigonometric Functions Using a Point on the Unit Circle
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
In the problem, t is a real number and P = (x, y) is the point on the unit circle that corresponds to t. Find the exact value
of the indicated trigonometric function of t.
3
55
1) ( ,
)
Find sin t.
8
8
55
8
A)
4
65
2) ( ,
)
9 9
A)
3) (
4) (-
65 4
, )
9
9
A)
4
9
C)
3 55
55
D)
3
8
B)
65
4
C)
9
4
D)
65
9
C)
3 55
55
D)
8 55
55
Find sec t.
55
3
A)
55
3
Find tan t.
4 65
65
55 3
, )
8
8
B)
B)
8
3
Find cos t.
B) -
65
9
C) -
Page 13
65
4
D) -
9 65
65
5) (-
39 5
, )
8
8
A)
6) (-
5
8
B) -
8
5
39
8
D) -
39
5
5
6
D) -
6 11
11
C)
33
4
D) -
4 33
33
C)
11
5
D) -
11
6
5
8
D) -
39
8
D) -
7 10
20
C)
11
5
, - ) Find sin t.
6
6
11
6
A) -
7) (-
Find cot t.
B)
6
5
C) -
33
4
, - ) Find cot t.
7
7
A)
33
7
5
11
8) ( , -
)
6
6
A) -
33
4
Find csc t.
6 11
11
5
39
9) ( , -
)
8
8
A)
B) -
11
6
B)
Find cos t.
39
8
B)
5
8
C) -
B)
7
3
C)
3
2 10
10) ( , -
) Find csc t.
7
7
A) -
10
6
3
7
2 Find the Exact Values of the Trigonometric Functions of Quadrantal Angles
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value. Do not use a calculator.
1) sin 2π
A) 0
2
2
C) 1
D) undefined
B) 0
C) 1
D) undefined
B) 0
C)
B)
2) cos 0
A)
2
2
3) tan 0
A) 1
Page 14
2
2
D) undefined
4) cot 0
A) 0
2
2
D) undefined
B) 1
C)
A) 1
B) 0
C) -1
D) undefined
6) tan π
A) 1
B) 0
C) -1
D) undefined
7) cos π
A) 0
B) -1
C) 1
D) undefined
B) -1
C) 0
D) undefined
B) 0
C) 1
D) undefined
A) -1
B) 0
C) 1
D) undefined
11) cos (-π)
A) 1
B) 0
C) -1
D) undefined
5) cot
π
2
8) cot
3π
2
A) 1
9) sin (38π)
A) -1
π
10) sin (- )
2
3 Find the Exact Values of the Trigonometric Functions of π/4 = 45°
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value. Do not use a calculator.
π
1) cos
4
A)
2
B) -
2
2
C)
3
2
D)
B)
2
C)
2
2
D)
2
2
2) cos 45°
A)
3
2
Find the exact value of the expression if θ = 45°. Do not use a calculator.
3) f(θ) = sec θ
Find f(θ).
2 3
B) - 2
C) 2
A)
3
4) g(θ) = sin θ
1
A)
2
1
2
D)
Find [g(θ)]2 .
B)
C) -
2
Page 15
2
2
D) 2
2
2
5) f(θ) = cos θ
2
A)
2
Find 3f(θ).
6) g(θ) = sin θ
A) -6 2
Find 6g(θ).
B)
3 2
2
C) -
B) 6 2
3 2
2
C) 3 2
D) -
2
2
D) -3 2
Solve the problem.
7) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the
formula
2a
t =
g sinθ cosθ
where a is the length (in feet) of the base and g ≈ 32 feet per second per second is the acceleration of gravity.
How long does it take a block to slide down an inclined plane with base a = 12 when θ = 45°? If necessary,
round the answer to the nearest tenth of a second.
A) 1.5 sec
B) 1.3 sec
C) 0.3 sec
D) 1.2 sec
8) The force acting on a pendulum to bring it to its perpendicular resting point is called the restoring force. The
restoring force F, in Newtons, acting on a string pendulum is given by the formula
F = mg sinθ
where m is the mass in kilograms of the pendulumʹs bob, g ≈ 9.8 meters per second per second is the
acceleration due to gravity, and θ is angle at which the pendulum is displaced from the perpendicular. What is
the value of the restoring force when m = 0.9 kilogram and θ = 45°? If necessary, round the answer to the
nearest tenth of a Newton.
A) 6 N
B) 6.4 N
C) 7.5 N
D) 6.2 N
4 Find the Exact Values of the Trigonometric Functions of π/6 = 30° and π/3 = 60°
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value. Do not use a calculator.
1) cot 30°
3
3
A)
B)
2
3
C) 1
D)
3
2) csc 60°
A)
3) csc
B)
2
B)
2
C)
2 3
3
D) 2
π
6
A)
4) cot
3
2
1
2
C) 2
D)
2 3
3
D)
1
2
π
3
A) 1
B)
3
3
C)
Page 16
3
Find the exact value of the expression. Do not use a calculator.
5) cot 45° - cos 30°
3
2 3 - 3 2
B) -
A)
6
6
6) cot 60° - cos 45°
2 2 - 3 3
A)
6
7) cos 60° + tan 60°
3 3
A)
2
C)
2 - 2
2
D)
2 - 3
2
2 3 - 3 2
6
D)
2 - 3
2
D)
1 + 3
2
B)
2 - 2
2
C)
B)
1 + 2 3
2
C) 2 3
π
π
8) sin - cos
3
6
A)
3
B) 1
3 - 1
2
C)
D) 0
π
π
9) tan - cos
6
6
A) -
6
2
B)
2 3 - 3 2
6
C) -
3
6
D)
3
D)
3
3
1
2
D)
3
2
C) 1
D)
1
2
C) 6
D) -
3
2
11 3
2
C) -
D) -
3
2
Find the exact value of the expression if θ = 30°. Do not use a calculator.
10) f(θ) = sin θ
Find f(θ).
2
3
1
B)
C)
A)
2
2
2
11) g(θ) = cos θ
Find g(2θ).
A) 1
B)
12) f(θ) = sin θ
1
A)
4
Find [f(θ)]2 .
13) g(θ) = sin θ
Find 12g(θ).
B)
C)
3
4
B) -
A) 6 3
14) f(θ) = cos θ
11
A)
2
3
1
2
Find 11f(θ).
B)
1
2
Find the exact value of the expression if θ = 60°. Do not use a calculator.
15) f(θ) = csc θ
Find f(θ).
3
2 3
A) 2
B)
C)
2
3
Page 17
D) 2
16) g(θ) = cos θ
3
A)
4
Find [g(θ)]2 .
17) f(θ) = sin θ
1
A) -
2
Find 10f(θ).
18) g(θ) = cos θ
7 3
A)
2
Find 7g(θ).
B)
3
C)
3
2
B) 5 3
C) -
7
2
C) -
B)
3
2
1
2
D)
1
4
D) 5
D) -
3
2
Solve the problem.
19) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the
formula
2a
t =
g sinθ cosθ
where a is the length (in feet) of the base and g ≈ 32 feet per second per second is the acceleration of gravity.
How long does it take a block to slide down an inclined plane with base a = 15 when θ = 30°? If necessary,
round the answer to the nearest tenth of a second.
A) 1.2 sec
B) 1.5 sec
C) 2.5 sec
D) 0.4 sec
20) The force acting on a pendulum to bring it to its perpendicular resting point is called the restoring force. The
restoring force F, in Newtons, acting on a string pendulum is given by the formula
F = mg sinθ
where m is the mass in kilograms of the pendulumʹs bob, g ≈ 9.8 meters per second per second is the
acceleration due to gravity, and θ is angle at which the pendulum is displaced from the perpendicular. What is
the value of the restoring force when m = 0.5 kilogram and θ = 30°? If necessary, round the answer to the
nearest tenth of a Newton.
A) 4.8 N
B) 2.4 N
C) 4.2 N
D) 2.5 N
5 Find the Exact Values for Integer Multiples of π/6 = 30°, π/4 = 45°, and π/3 = 60°
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value. Do not use a calculator.
8π
1) cos
3
A)
2) sec
1
2
3
2
C)
3
2
D) -
1
2
B) - 2
C)
2
2
D) -
2 3
3
D) -
2
2
B) -
21π
4
A) -2
3) sin 405°
1
A) -
2
B)
2
2
C)
Page 18
1
2
4) cot 390°
3
A)
3
B)
3
3
C) -
3
D) - 3
Find the exact value of the expression. Do not use a calculator.
7π
5π
5) tan
+ tan
4
4
A)
2 2 + 1
6
6) sin 135° - sin 270°
2
A)
2
7) cos
B)
1
2
C) 0
D)
2 + 1
2
D)
2 - 2
2
D)
3 + 1
2
3
2
D) -
1
4
2 3 + 3
6
D) -
5 3
6
B)
2 + 2
2
C) 2
B)
3 + 3
3
C)
π
5π
+ tan
3
3
A)
2 3 + 3
6
8) cos 120° tan 60°
3
A)
2
9) tan 150° cos 210°
3 + 1
A)
2
10) sin 330° sin 270°
1
A)
2
1 - 2 3
2
B)
3
2
C) -
B)
3 3 + 2 3
6
C)
B) -
3
2
C) -
1
2
D)
3
2
6 Use a Calculator to Approximate the Value of a Trigonometric Function
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a calculator to find the approximate value of the expression rounded to two decimal places.
1) sin 17°
A) 0.21
B) -1.04
C) -0.96
D) 0.29
2) cos 39°
A) 0.15
B) 0.78
C) 0.27
D) 0.90
3) tan 82°
A) 0.33
B) 0.38
C) 7.17
D) 7.12
B) 1.00
C) 1.07
D) -0.24
4) cos
3π
5
A) -0.31
Page 19
5) sec
π
12
A) 1.04
B) 1.13
C) 0.91
D) 1.00
B) 1.09
C) 1.20
D) -37.55
A) 2.30
B) 2.41
C) 145.79
D) 145.90
8) cot 0.2944
A) 0.30
B) 3.30
C) 0.96
D) 1.04
9) cos 2
A) 1.00
B) -0.42
C) -1.00
D) 0.42
10) cos 1°
A) 0.54
B) 1.00
C) -0.54
D) -1.00
11) tan 37°
A) -0.84
B) 0.75
C) 0.80
D) 0.60
6) csc 66°
A) -37.66
7) cot
π
8
Solve the problem.
12) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by the
formula
2a
t =
g sinθ cosθ
where a is the length (in feet) of the base and g ≈ 32 feet per second per second is the acceleration of gravity.
How long does it take a block to slide down an inclined plane with base a = 10 when θ = 54°? If necessary,
round the answer to the nearest tenth of a second.
A) 1.1 sec
B) 1 sec
C) 0.3 sec
D) 1.2 sec
13) The force acting on a pendulum to bring it to its perpendicular resting point is called the restoring force. The
restoring force F, in Newtons, acting on a string pendulum is given by the formula
F = mg sinθ
where m is the mass in kilograms of the pendulumʹs bob, g ≈ 9.8 meters per second per second is the
acceleration due to gravity, and θ is angle at which the pendulum is displaced from the perpendicular. What is
the value of the restoring force when m = 0.7 kilogram and θ = 83°? If necessary, round the answer to the
nearest tenth of a Newton.
A) 6.8 N
B) 7 N
C) 6.6 N
D) 0.8 N
Page 20
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) The strength S of a wooden beam with rectangular cross section is given by the formula
S = kd3 sin2 θ cos θ
where d is the diagonal length, θ the angle illustrated, and k is a constant that varies with the type of wood
used.
Let d = 1 and express the strength S in terms of the constant k for θ = 45°, 50°, 55°, 60°, and 65°. Does the
strength always increase as θ gets larger?
7 Use a Circle of Radius r to Evaluate the Trigonometric Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
A point on the terminal side of an angle θ is given. Find the exact value of the indicated trigonometric function of θ.
Find sin θ.
1) (-3, -4)
3
4
4
3
B)
C)
D) -
A) -
5
5
5
5
2) (-3, 4)
3
A)
5
Find cos θ.
1 1
3) (- , )
2 3
A)
A)
C) -
4
5
D)
4
5
B) -
13
2
C) -
3 13
13
D)
13
3
13
2
C)
3
2
D) -
C)
3
2
D)
13
2
C) -2
D)
2
C) - 26
D) -
Find tan θ.
2
3
B)
5) (2, 3)
13
2
Find cot θ.
2
A)
3
13
2
B) -
6) (-1, -1)
A) - 2
Find csc θ.
7) (-5, -1)
Find sec θ.
A)
3
5
Find cos θ.
2 13
13
4) (2, 3)
B) -
26
5
B) -1
B) -
3 26
26
Page 21
26
5
Solve the problem.
8) If sin θ = 0.2, find sin (θ + π).
A) -0.2
C) -0.8
B) 0.2
D) 0.8
1
9) If sin θ = , find csc θ.
7
A) 7
B)
6
7
C) -
1
7
D) undefined
10) A racetrack curve is banked so that the outside of the curve is slightly elevated or inclined above the inside of
the curve. This inclination is called the elevation of the track. The maximum speed on the track in miles per
hour is given by
r(29000 + 41000 tan θ)
where r is the radius of the track in miles and θ is the elevation in degrees. Find the maximum speed for a
racetrack with an elevation of 29° and a radius of 0.6 miles. Round to the nearest mile per hour.
A) 50,638 mph
B) 176 mph
C) 200 mph
D) 40,067 mph
11) The path of a projectile fired at an inclination θ to the horizontal with an initial speed vo is a parabola. The
range R of the projectile, the horizontal distance that the projectile travels, is found by the formula
vo2 sin 2θ
where g = 32.2 feet per second per second or g = 9.8 meters per second per second. Find the
R =
g
range of a projectile fired with an initial velocity of 197 feet per second at an angle of 17° to the horizontal.
Round your answer to two decimal places.
A) 704.76 ft
B) 673.97 ft
C) 673.87 ft
D) 352.38 ft
2.3 Properties of the Trigonometric Functions
1 Determine the Domain and the Range of the Trigonometric Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) What is the domain of the cosine function?
A) all real numbers
B) all real numbers, except integral multiples of π (180°)
C) all real numbers from -1 to 1, inclusive
π
D) all real numbers, except odd multiples of (90°)
2
2) For what numbers θ is f(θ) = sec θ not defined?
A) all real numbers
B) integral multiples of π (180°)
C) odd multiples of π (180°)
D) odd multiples of
π
(90°)
2
3) For what numbers θ is f(θ) = csc θ not defined?
A) all real numbers
C) odd multiples of
B) integral multiples of π (180°)
π
(90°)
2
D) odd multiples of π (180°)
Page 22
4) What is the range of the cosine function?
A) all real numbers from -1 to 1, inclusive
B) all real numbers
C) all real numbers greater than or equal to 0
D) all real numbers greater than or equal to 1 or less than or equal to -1
5) What is the range of the cotangent function?
A) all real numbers greater than or equal to 1 or less than or equal to -1
B) all real numbers
C) all real numbers from -1 to 1, inclusive
D) all real numbers, except integral multiples of π(180)°
6) What is the range of the cosecant function?
A) all real numbers greater than or equal to 1 or less than or equal to -1
B) all real numbers
C) all real numbers from -1 to 1, inclusive
D) all real numbers, except integral multiples of π(180)°
2 Determine the Period of the Trigonometric Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the fact that the trigonometric functions are periodic to find the exact value of the expression. Do not use a
calculator.
1) sin 495°
2
1
2
1
B) -
C) -
D)
A)
2
2
2
2
2) tan 390°
A)
3
B)
3
3
C) - 3
D)
3
2
3) csc 660°
A) - 3
B) - 2
C) -
2 3
3
D) -
1
2
4) cot 750°
3
B) -
5) cot 720°
A) 3
A)
3
3
3
3
C) - 3
D)
B) -1
C) 0
D) undefined
B) 0
C)
6) tan 720°
A) 1
7) cos
3
3
D) undefined
20π
3
A)
1
2
B) -
3
2
C) -
Page 23
1
2
D)
3
2
8) sin
22π
3
A) -
9) tan
B) -1
3
B)
3
2
C) -
3
2
D)
9π
4
A)
10) sec
1
2
3
3
C) -1
D) 1
21π
4
A)
2
2
B) -2
C) -
2 3
3
Solve the problem.
11) If cos θ = -0.8, find the value of cos θ + cos (θ + 2π) + cos (θ + 4π).
A) -0.8
B) -2.4 + 6π
C) -2.4
12) If tan θ = 2.9, find the value of tan θ + tan (θ + π) + tan (θ + 2π).
A) 8.7 + 3π
B) 10.7
C) 8.7
D) - 2
D) -0.4
D) undefined
1
13) If f(θ) = sin θ and f(a) = , find the exact value of f(a) + f(a + 2π) + f(a + 4π).
6
A)
1
6
B)
1
+ 6π
2
C)
5
2
D)
14) If f(θ) = cot θ and f(a) = -4, find the exact value of f(a) + f(a + π) + f(a + 3π).
A) -4
B) -12
C) -12 + 4π
15) If f(θ) = cos θ and f(a) = -
A) -12
1
2
D) undefined
1
, find the exact value of f(a) + f(a - 2π) + f(a + 4π).
12
B) -36
C) -
1
4
D) -
1
12
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
1
16) If f(θ) = sin θ and f(a) = - , find the exact value of f(a) + f(a - 4π) + f(a - 2π).
9
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
17) If sin θ = -0.8, find the value of sin θ + sin (θ + 2π) + sin (θ + 4π).
A) -0.4
B) -2.4 + 6π
C) -2.4
D) -0.8
18) If cot θ = 7.3, find the value of cot θ + cot (θ + π) + cot (θ + 2π).
A) 21.9
B) 23.9
C) 21.9 + 3π
D) undefined
Page 24
3 Determine the Signs of the Trigonometric Functions in a Given Quadrant
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Name the quadrant in which the angle θ lies.
1) tan θ > 0, sin θ < 0
A) I
B) II
2) cos θ < 0,
A) I
csc θ < 0
3) sin θ > 0,
A) I
cos θ < 0
4) cot θ < 0,
A) I
cos θ > 0
5) csc θ > 0,
A) I
sec θ > 0
6) sec θ < 0,
A) I
tan θ < 0
7) tan θ < 0,
A) I
sin θ < 0
8) cos θ > 0,
A) I
csc θ < 0
9) cot θ > 0,
A) I
sin θ < 0
10) sin θ > 0,
A) I
cos θ > 0
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
B) II
C) III
D) IV
C) II, III, and IV
D) III only
Solve the problem.
11) Which of the following trigonometric values are negative?
I. sin(-292°)
II. tan(-193°)
III. cos(-207°)
IV. cot 222°
A) II and III
B) I and III
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12) Determine the sign of the trigonometric values listed below.
(i) sin 250°
(ii) tan 330°
(iii) cos(-40°)
Page 25
