Ch. 7 Trigonometric Functions
7.1 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)
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3) 2π
3
A) B)
C) D)
4) – 3π
4
A) B)
C) D)
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5) –150°
A) B)
C) D)
6) 330°
A) B)
C) D)
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7) – 7π
6
A) B)
C) D)
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8) 5π
3
A) B)
C) D)
9) –120°
A) B)
C) D)
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10) 7π
4
A) B)
C) D)
Convert the angle to a decimal in degrees. Round the answer to two decimal places.
11) 88°32′52”
A) 88.55° B) 88.56° C) 88.51° D) 88.61°
12) 205°6‘57”
A) 205.12° B) 205.13° C) 205.08° D) 205.18°
13) 239°29′50”
A) 239.50° B) 239.51° C) 239.46° D) 239.56°
14) 23°47’37”
A) 23.79° B) 23.84° C) 23.52° D) 23.94°
15) 21°17’34”
A) 21.29° B) 21.34° C) 21.22° D) 21.37°
Convert the angle to D° M’ S” form. Round the answer to the nearest second.
16) 98.51°
A) 98°30′36” B) 98°30′42” C) 98°30′24” D) 98°30’51”
17) 135.87°
A) 135°52′12” B) 135°53′12” C) 135°50′87” D) 135°52’87”
18) 351.26°
A) 351°15′36” B) 351°15′26” C) 351°35′26” D) 351°16’35”
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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 = 23.24 centimeters, θ = 5.8 radians, s =?
A) 134.8 cm B) 135.8 cm C) 133.8 cm D) 136.8 cm
2) r = 10.2 inches, θ = 315°
,
s = ?
A) 56.1 in. B) 56.2 in. C) 56.3 in. D) 56.4 in.
3) r = 1
2 feet, s = 6 feet, θ = ?
A) 12 radians B) 3 radians C) 12° D) 3°
4) s = 1.82 meters, θ = 1.4 radians, r = ?
A) 1.3 m B) 0.77 m C) 0.5 m D) 0.65 m
Find the length s. Round the answer to three decimal places.
5)
s
π
4
3 m
A) 2.356 m B) 4.189 m C) 4.712 m D) 3.82 m
6)
π
6s
3 yd
A) 1.571 yd B) 6.283 yd C) 3.142 yd D) 5.73 yd
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7)
s
75°
7 yd
A) 9.163 yd B) 10.079 yd C) 8.247 yd D) 7.33 yd
8)
s
35°
2 m
A) 1.222 m B) 1.344 m C) 1.1 m D) 0.978 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) 376.99 ft B) 6.28 ft C) 2.09 ft D) 4.19 ft
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) 4.19 ft B) 4.25 ft C) 4.35 ft D) 4.40 ft
11) A ship in the Pacific Ocean measures its position to be 35°51′ north latitude. Another ship is reported to be
due north of the first ship at 43°16′ 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) 513 mi B) 29,370 mi C) 528 mi D) 29,385 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°45′
north latitude) and Flagstaff (35°16′ 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) 8.79 in. C) 7.51 in. D) 4.54 in.
14) A pendulum swings though an angle of 50° each second. If the pendulum is 55 inches long, how far does
its tip move each second? If necessary, round the answer to two decimal places.
A) 48 in. B) 50.43 in. C) 49.29 in. D) 46.15 in.
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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) 30°
A) π
6B) π
7C) π
8D) π
5
2) –90°
A) – π
2B) – π
3C) – π
4D) – π
8
3) 162°
A) 9π
10 B) 10π
11 C) 8π
9D) 10π
9
4) –480°
A) – 8π
3B) – 9π
4C) – 7π
2D) – 3π
8
5) 87°
A) 29π
60 B) 29π
90 C) 29π
30 D) 29π
120
6) 6°
A) π
30 B) π
60 C) π
15 D) π
18
Convert the angle in radians to degrees.
7) 2π
7
A) 51.43° B) 52.43° C) 53.43° D) 50.43°
8) – 5π
6
A) –150° B) –151° C) –152° D) –149°
9) π
4
A) 45° B) 1° C) 4° D) 45π°
10) – π
5
A) –36° B) –1° C) 1° D) –36π°
11) 5π
4
A) 225° B) 450° C) 144° D) 144π°
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12) – 17
6π
A) –510° B) –1020π°C)
–9° D) –255°
13) π
6
A) 30° B) 60° C) 1080° D) 15°
14) 11π
12
A) 165° B) 160° C) 150° D) 210°
Convert the angle in degrees to radians. Express the answer in decimal form, rounded to two decimal places.
15) 79°
A) 1.38 B) 1.37 C) 1.36 D) 1.35
16) –124°
A) –2.16 B) –2.15 C) –2.14 D) –2.13
Convert the angle in radians to degrees. Express the answer in decimal form, rounded to two decimal places.
17) 5
A) 286.48° B) 286.96° C) 0.09° D) 0.24°
18) 3.67
A) 210.28° B) 211.05° C) 0.06° D) 0.21°
19) 2
A) 81.03° B) 81.28° C) 0.02° D) 0.08°
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.
1) r = 11 inches
,
θ = 4 radians, A = ?
A) 242 in2B) 484 in2C) 22 in2D) 44 in2
2) r = 17 feet, A = 39 square feet, θ = ?
A) 0.27 radians B) 0.13 radians C) 5635.5 radians D) 11,271 radians
3) θ = π
4 radians, A = 79 square meters, r = ?
A) 14.19 m B) 124.03 m C) 31.01 m D) 5.57 m
4) r = 8 inches, θ = 90°
,
A = ?
A) 50.24 in2B) 100.48 in2C) 6.28 in2D) 12.56 in2
5) r = 10 feet, A = 100 square feet, θ = ?
A) 114.65° B) 57.32° C) 286,624.2° D) 573,248.41°
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6) θ = 60°
,
A = 57 square meters, r = ?
A) 10.44 m B) 119.32 m C) 29.83 m D) 5.46 m
7) r = 76.9 centimeters, θ = π
12 radians, A = ?
A) 774.1 cm2B) 10.1 cm2C) 246.4 cm2D) 1548.2 cm2
8) r = 34.6 feet, θ = 2.266°
,
A = ?
A) 23.67 ft2B) 26.67 ft2C) 47.34 ft2D) 50.34 ft2
Find the area A. Round the answer to three decimal places.
9)
π
3
12 m
A) 75.398 m2B) 6.283 m2C) 150.796 m2D) 48 m2
10)
π
5
4 yd
A) 5.027 yd2B) 1.257 yd2C) 10.053 yd2D) 3.2 yd2
11)
55°
3 yd
A) 4.32 yd2B) 1.44 yd2C) 8.639 yd2D) 1.375 yd2
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12)
25°
4 yd
A) 3.491 yd2B) 0.873 yd2C) 6.981 yd2D) 1.111 yd2
Solve the problem.
13) A circle has a radius of 7 centimeters. Find the area of the sector of the circle formed by an angle of 70°. If
necessary, round the answer to two decimal places.
A) 29.93 cm2B) 4.28 cm2C) 59.86 cm2D) 9.53 cm2
14) An irrigation sprinkler in a field of lettuce sprays water over a distance of 40 feet as it rotates through an
angle of 150°. What area of the field receives water? If necessary, round the answer to two decimal places.
A) 2094.4 ft2B) 52.36 ft2C) 4188.79 ft2D) 666.67 ft2
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) 3.5° B) 1.75° C) 2.63° D) 11°
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) 1036.7 in2B) 1105.3 in2C) 1041.8 in2D) 948.3 in2
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 1
3
radian is swept out, what is the linear speed of the object?
A) 1
6 cm/sec B) 6 cm/sec C) 1
6 radians/sec D) 6 radians/sec
2) An object is traveling around a circle with a radius of 20 meters. If in 10 seconds a central angle of 1
5 radian
is swept out, what is the linear speed of the object?
A) 2
5 m/sec B) 1
5 m/sec C) 1
4 m/sec D) 1
8 m/sec
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?
A) 2 m/sec B) 3 m/sec C) 2
3 m/sec D) 1
3 m/sec
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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) 8.7 feet C) 9.0 feet D) 8.1 feet
5) A gear with a radius of 8 centimeters is turning at π
11 radians/sec. What is the linear speed at a point on the
outer edge of the gear?
A) 8π
11 cm/sec B) 11π
8 cm/sec C) 88πcm/sec D) π
88 cm/sec
6) A wheel of radius 5.0 feet is moving forward at 16 feet per second. How fast is the wheel rotating?
A) 3.2 radians/sec B) 0.31 radians/sec C) 5.1 radians/sec D) 0.63 radians/sec
7) A car is traveling at 35 mph. If its tires have a diameter of 27 inches, how fast are the car’s tires turning?
Express the answer in revolutions per minute. If necessary, round to two decimal places.
A) 435.73 rpm B) 2737.78 rpm C) 450.73 rpm D) 871.46 rpm
8) A pick–up truck is fitted with new tires which have a diameter of 41 inches. How fast will the pick–up
truck be moving when the wheels are rotating at 380 revolutions per minute? Express the answer in miles
per hour rounded to the nearest whole number.
A) 46 mph B) 7 mph C) 53 mph D) 23 mph
9) The Earth rotates about its pole once every 24 hours. The distance from the pole to a location on Earth 24°
north latitude is about 3617.6 miles. Therefore, a location on Earth at 24° north latitude is spinning on a
circle of radius 3617.6 miles. Compute the linear speed on the surface of the Earth at 24° north latitude.
A) 947 mph B) 151 mph C) 22,730 mph D) 1002 mph
10) To approximate the speed of a river, a circular paddle wheel with radius 0.43 feet is lowered into the
water. If the current causes the wheel to rotate at a speed of 9 revolutions per minute, what is the speed of
the current? If necessary, round to two decimal places.
A) 0.28 mph B) 0.04 mph C) 24.32 mph D) 0.14 mph
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)
Io 4.214 × 105 42.460
Europa 6.709 × 105 85.243
Ganymeade 1.070 × 106171.709
Callisto 1.883 × 106 400.536
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 Earth’s surface of 2.8 x 104
miles. The orbit is circular, and one revolution around Earth takes 10.9 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) 768 mph B) 16,504 mph C) 122 mph D) 673 mph
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13) A carousel has a radius of 15 feet and takes 29 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) 3.25 ft/sec B) 0.52 ft/sec C) 12.15 ft/sec D) 94.25 ft/sec
7.2 Right Triangle Trigonometry
1 Find the Values of Trigonometric Functions of Acute Angles
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the
given angle. Give exact answers with rational denominators.
1) Find sin A when a = 5 and b = 7.
A) 574
74 B) 774
74 C) 74
5D) 74
7
2) Find sin B when b = 3 and c = 10.
A) 3
10 B) 10 91
91 C) 391
91 D) 91
10
3) Find cos A when a = 8 and b = 5.
A) 589
89 B) 89
8C) 89
5D) 889
89
4) Find cos A when a = 10 and c = 11.
A) 111
11 B) 10
11 C) 10
11 D) 111
10
5) Find csc B when a = 7 and b = 4.
A) 65
4B) 65
7C) 465
65 D) 765
65
6) Find sec B when a = 2 and b = 3.
A) 13
2B) 213
13 C) 213
3D) 313
13
7) Find tan A when a = 3 and b = 2.
A) 3
2B) 2
3C) 13
2D) 13
3
8) Find tan B when a = 4 and b = 9.
A) 9
4B) 997
97 C) 4
9D) 497
97
9) Find cot A when a = 9 and c = 10.
A) 19
9B) 19
10 C) 919
19 D) 10 19
19
10) Find cot A when b = 5 and c = 6.
A) 511
11 B) 611
11 C) 11
5D) 11
6
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Solve the problem.
11) Find tan P.
A) tan P = 3
5B) tan P = 4
5C) tan P = 4
3D) tan P = 3
4
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12) Find the exact value of each of the six trigonometric functions of the angle P.
2 Use the Fundamental Identities
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use identities to find the exact value of the indicated trigonometric function of the acute angle θ.
1) sin θ = 5
3, cos θ = 2
3Find tan θ.
A) 5
2B) 25
5C) 3
2D) 35
5
2) sin θ = 7
4, cos θ = 3
4Find cot θ.
A) 37
7B) 7
3C) 4
3D) 47
7
3) sin θ = 22
3, cos θ = 1
3Find sec θ.
A) 3 B) 2 2 C) 2
4D) 32
4
4) sin θ = 7
4, cos θ = 3
4Find csc θ.
A) 47
7B) 7
3C) 37
7D) 4
3
Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of the expression.
Do not use a calculator.
5) sin2 70° + cos2 70°
A) 1 B) 0 C) 2 D) –1
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6) sec2 20° – tan2 20°
A) 1 B) 0 C) –1D)2
7) sin 35° csc 35°
A) 1 B) 0 C) 35 D) –1
8) tan 70° – sin 70°
cos 70°
A) 0 B) 1 C) 70 D) undefined
3 Find the Values of the Remaining Trigonometric Functions, Given the Value of One of Them
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the definition or identities to find the exact value of the indicated trigonometric function of the acute angle θ.
1) sin θ = 4
5Find tan θ.
A) 4
3B) 3
4C) 5
3D) 5
4
2) sin θ = 5
13 Find csc θ.
A) 13
5B) 5
12 C) 13
12 D) 12
5
3) cos θ = 3
5Find sec θ.
A) 5
3B) 5
4C) 4
3D) 3
4
4) cos θ = 4
5Find cot θ.
A) 4
3B) 3
4C) 5
3D) 5
4
5) tan θ = 3 Find sin θ.
A) 310
10 B) 10
10 C) 10
3D) 10
6) tan θ = 3 Find cos θ.
A) 10
10 B) 310
10 C) 1
3D) 10
7) cot θ = 1
3Find sin θ.
A) 310
10 B) 10
10 C) 3 D) 10
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8) sec θ = 5
3Find csc θ.
A) 5
4B) 4
5C) 3
5D) 3
4
9) sec θ = 10 Find cot θ.
A) 1
3B) 310
10 C) 10
10 D) 10
3
10) csc θ = 23
3Find cos θ.
A) 1
2B) 3
2C) 3
3D) 2
11) tan θ = 7
15 Find sin θ and cos θ.
A) sin θ = 7
8, cos θ = 15
8B) sin θ = 8
7, cos θ = 15
8
C) sin θ = 15
8, cos θ = 7
8D) sin θ = 7
8, cos θ = 815
15
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12) cos θ = 26
5Find sin θ and tan θ.
13) tan θ = 26 Find the remaining five trigonometric functions of the acute angle θ.
4 Use the Complementary Angle Theorem
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use Fundamental Identities and/or the Complementary Angle Theorem to find the exact value of the expression.
Do not use a calculator.
1) – sec 10°
csc 80°
A) –1 B) 0 C) 1 D) undefined
2) csc2 5° – tan2 85°
A) 1 B) 0 C) –1D)2
3) tan 85° – cos 5°
cos 85°
A) 0 B) 1 C) –1D)2
4) cos 15°sin 75° + sin 15°cos 75°
A) 1 B) 0 C) –1D)2
5) If tan2 θ = 11, find the exact value of sec2 θ.
A) 12 B) 11 C) 10 D) 22
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6) If tan θ = 2, find the exact value of cot π
2 – θ.
A) 2 B) 1
2C) 1 D) 3
Solve the problem.
7) Given sin 30° = 1
2 , use trigonometric identities to find the exact value of tan π
6.
A) 3
3B) 3 C) 3 D) 3 3
8) Given cos 30° = 3
2 , use trigonometric identities to find the exact value of csc π
3.
A) 23
3B) 3 C) 2
3D) 3
3
9) Given cot θ = 4 , use trigonometric identities to find the exact value of csc2 θ.
A) 17 B) 17 C) 1
4D) 17
4
10) Given csc θ = 2 , use trigonometric identities to find the exact value of sec2 θ.
A) 4
3B) 1
4C) 1
3D) 1
2
11) Given tan θ = 2, use trigonometric identities to find the exact value of csc π
2 – θ.
A) 5 B) 2 C) 1
2D) 5
2
12) Given the approximation sin 30° ≈ 0.50
,
use trigonometric identities to find the approximate value of cos
30°. Round the answer to two decimal places.
A) 0.87 B) 0.50 C) 0.58 D) 1.15
13) Given the approximation sin 28° ≈ 0.47
,
use trigonometric identities to find the approximate value of cot
28°. If necessary, round the answer to two decimal places.
A) 1.88 B) 0.53 C) 0.88 D) 1.13
14) Given the approximation cos 39° ≈ 0.78
,
use trigonometric identities to find the approximate value o
f
csc
39°. If necessary, round the answer to two decimal places.
A) 1.59 B) 1.29 C) 0.63 D) 0.81
15) Given the approximation cos 25° ≈ 0.91
,
use trigonometric identities to find the approximate value o
f
sin
65°. If necessary, round the answer to two decimal places.
A) 0.91 B) 0.42 C) 0.47 D) 1.1
16) If csc θ = 2, find the value of csc θ + sec(90° –θ). If necessary, round the answer to two decimal places.
A) 1.58 B) 2 C) –0.42 D) 4
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SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
17) If cos θ = 1
6, find the exact value of (i) sin(90° – θ) and (ii) csc ( π
2 – θ).
18) A barge is located 200 feet away from the coastline 1200 feet down the coast from a power source at point
A. To supply the barge with electricity, a power line will be run from point A to a point B on the coast and
then from point B to the barge. If power lines on land coast $3 per foot and power lines under water cost
$5 per foot, calculate the total cost of running a power line from point A to the barge when point B is 800
feet down the coast from point A in the direction of the barge. Recalculate the total cost when the distance
from A to B is increased in 50–foot increments, until you locate a possible minimum cost. Interpret your
solution.
7.3 Computing the Values of Trigonometric Functions of Acute Angles
1 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.
1) sin π
4
A) 2
2B) –2
2C) 2 D) 1
2
2) csc 45°
A) 2 B) 2
2C) 3
2D) 3
Find the exact value of the expression if θ = 45°. Do not use a calculator.
3) f(θ) = sec θFind f(θ).
A) 2 B) –2 C) 2
2D) 23
3
4) g(θ) = cos θFind [g(θ)]2.
A) 1
2B) 2 C) – 2
2D) 2
5) f(θ) = sin θFind 9 f(θ).
A) 92
2B) – 92
2C) – 2
2D) 2
2
6) g(θ) = sin θFind 8 g(θ).
A) 4 2 B) 8 2 C) –42 D) –82
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7) f(θ) = cos θFind f(θ)
5.
A) 2
10 B) 2
5C) 52
2D) 10 2
2 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) sec 30°
A) 23
3B) 2 C) 3
2D) 2
2) tan 60°
A) 3 B) 3
3C) 3
2D) 2
3) cos π
6
A) 3
2B) 3 C) 23
3D) 2
2
4) cos π
3
A) 1
2B) 3
2C) 23
3D) 2
2
Find the exact value of the expression if θ = 30°. Do not use a calculator.
5) f(θ) = tan θFind f(θ).
A) 3
3B) 3 C) 3
2D) 1
6) g(θ) = cos θFind g(2θ).
A) 1
2B) 3
2C) 1 D) 3
7) f(θ) = cos θFind [f(θ)]2.
A) 3
4B) 3
2C) 3 D) 1
4
8) g(θ) = sin θFind 10 g(θ).
A) 5 B) 5 3 C) – 1
2D) – 3
2
9) f(θ) = cos θFind 3 f(θ).
A) 33
2B) 3
2C) – 1
2D) – 3
2
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10) g(θ) = sin θFind g(θ)
4.
A) 1
8B) 3
8C) 1
4D) 8 3
11) f(θ) = cos θFind f(θ)
4.
A) 3
8B) 3
2C) 8 3 D) 2 3
Find the exact value of the expression if θ = 60°. Do not use a calculator.
12) f(θ) = cot θFind f(θ).
A) 3
3B) 3 C) 1
2D) 1
13) g(θ) = sin θFind [g(θ)]2.
A) 3
4B) 1
2C) 1 D) 1
4
14) f(θ) = sin θFind 4 f(θ).
A) 2 3 B) 2 C) – 1
2D) – 3
2
15) g(θ) = cos θFind 9 g(θ).
A) 9
2B) 93
2C) – 1
2D) – 3
2
16) f(θ) = sin θFind f(θ)
3.
A) 3
6B) 1
6C) 3
3D) 6 3
17) g(θ) = cos θFind g(θ)
4.
A) 1
8B) 8 3 C) 3
8D) 3
2
Find the exact value of the expression. Do not use a calculator.
18) tan 45° – cos 45°
A) 2 – 2
2B) 23
– 32
6C) 2 – 3
2D) – 3
6
19) sec 30° – sin 45°
A) 43
– 32
6B) 4 – 2
2C) 4 – 3
2D) 42
– 33
6
20) cos 60° + tan 60°
A) 1 + 23
2B) 2 3 C) 1 + 3
2D) 33
2
Page 21
21) sin π
3 – cos π
6
A) 0 B) 3 C) 3 – 1
2D) 1
22) cot π
3 – cos π
6
A) – 3
6B) 3 C) – 6
2D) 23
– 32
6
23) 1 – sin2 30° – sin2 60°
A) 0 B) 1 C) 1
4D) 1 – 3
2
24) 1 + cot2 30° – sec2 45°
A) 2 B) 1 C) 0 D) 3
25) 3 cot2 45° + 8 sin2 30°
A) 5 B) πC) 3 D) 7
26) sin2 60° – cos2 45° – sin2 30°
A) 0 B) 3
2C) – 1
2D) 3
3 Use a Calculator to Approximate the Values of the Trigonometric Functions of Acute Angles
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. Round the answer to two decimal places.
1) sin 85°
A) 1.00 B) 0.93 C) –0.18 D) –0.11
2) cos 52°
A) 0.62 B) 0.71 C) –0.16 D) –0.07
3) tan 11°
A) 0.19 B) 0.30 C) –225.95 D) –226.06
4) cos 3π
5
A) –0.31 B) –0.37 C) 1.00 D) 1.06
5) sec π
7
A) 1.11 B) 1.16 C) 1.00 D) 0.95
6) csc 85°
A) 1.00 B) 1.06 C) –5.68 D) –5.62
Page 22
7) cot π
5
A) 1.38 B) 1.49 C) 91.18 D) 91.07
8) cot 0.1655
A) 5.99 B) 0.17 C) 0.99 D) 1.01
9) cos 7
A) 0.75 B) 0.99 C) –0.75 D) –0.99
10) cos 3°
A) 1.00 B) 0.99 C) –1.00 D) –0.99
11) tan 37°
A) 0.75 B) –0.84 C) 0.80 D) 0.60
4 Model and Solve Applied Problems Involving Right Triangles
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by
the formula
t = 2a
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) 1.2 sec B) 1.3 sec C) 0.3 sec D) 1.5 sec
2) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by
the formula
t = 2a
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 = 9 when θ = 30°? If
necessary, round the answer to the nearest tenth of a second.
A) 1.1 sec B) 1.9 sec C) 0.3 sec D) 1.4 sec
3) If friction is ignored, the time t (in seconds) required for a block to slide down an inclined plane is given by
the formula
t = 2a
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 = 14 when θ = 61°? If
necessary, round the answer to the nearest tenth of a second.
A) 1.4 sec B) 1.9 sec C) 0.4 sec D) 1.8 sec
Page 23
4) 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 pendulum’s 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 = 45°? If necessary, round the answer
to the nearest tenth of a Newton.
A) 4.9 N B) 5.8 N C) 4.7 N D) 4.8 N
5) 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 pendulum’s 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 = 0.6 kilogram and θ = 30°? If necessary, round the answer
to the nearest tenth of a Newton.
A) 2.9 N B) 5.8 N C) 5.1 N D) 2.6 N
6) 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 pendulum’s 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 = 0.6 kilogram and θ = 64°? If necessary, round the answer
to the nearest tenth of a Newton.
A) 5.3 N B) 5.4 N C) 2.6 N D) 5.5 N
7) A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the lake
80 feet from a piling that is directly across from a pier on the other side of the lake. From his transit, the
angle between the piling and the pier is 70°. What is the distance between the piling and the pier to the
nearest foot?
A) 220 ft B) 75 ft C) 27 ft D) 29 ft
8) A radio transmission tower is 120 feet tall. How long should a guy wire be if it is to be attached 8 feet from
the top and is to make an angle of 22° with the ground? Give your answer to the nearest tenth of a foot.
A) 299.0 ft B) 320.3 ft C) 120.8 ft D) 129.4 ft
9) A building 220 feet tall casts a 90 foot long shadow. If a person looks down from the top of the building,
what is the measure of the angle between the end of the shadow and the vertical side of the building (to
the nearest degree)? (Assume the person’s eyes are level with the top of the building.)
A) 22° B) 68° C) 66° D) 24°
10)
J
ohn (whose line of sight is 6 ft above horizontal) is trying to estimate the height of a tall oak tree. He first
measures the angle of elevation from where he is standing as 35°. He walks 30 feet closer to the tree and
finds that the angle of elevation has increased by 12°. Estimate the height of the tree rounded to the nearest
whole number.
A) 67 ft B) 90 ft C) 86 ft D) 61 ft
11) A photographer points a camera at a window in a nearby building forming an angle of 42° with the
camera platform. If the camera is 52 m from the building, how high above the platform is the window, to
the nearest hundredth of a meter?
A) 46.82 m B) 0.9 m C) 1.11 m D) 57.75 m
Page 24
12) A tree casts a shadow of 26 meters when the angle of elevation of the sun is 24°. Find the height of the tree
to the nearest meter.
A) 12 m B) 11 m C) 10 m D) 13 m
13) A twenty–five foot ladder just reaches the top of a house and forms an angle of 41.5° with the wall of the
house. How tall is the house? Round your answer to the nearest 0.1 foot.
A) 18.7 ft B) 18.6 ft C) 18.8 ft D) 19 ft
14) Two hikers on opposite sides of a canyon each stand precisely 525 meters above the canyon floor. They
each sight a landmark on the canyon floor on a line directly between them. The angles of depression from
each hiker to the landmark meter are 37° and 21°. How far apart are the hikers? Round your answer to the
nearest whole meter.
A) 2064 m B) 2065 m C) 2063 m D) 1064 m
15) From the edge of a 1000–foot cliff, the angles of depression to two cars in the valley below are 21° and 28°.
How far apart are the cars? Round your answers to the nearest 0.1 ft.
A) 724.4 ft B) 724.5 ft C) 714.4 ft D) 713.4 ft
7.4 Trigonometric Functions of Any Angle
1 Find the Exact Values of the Trigonometric Functions for Any Angle
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
A point on the terminal side of angle θ is given. Find the exact value of the indicated trigonometric function.
1) (6
,
8) Find sin θ.
A) 4
5B) 3
5C) 3
4D) 4
3
2) (15
,
20) Find cos θ.
A) 3
5B) 4
5C) 3
4D) 4
3
3) (–15
,
36) Find sin θ.
A) 12
13 B) 5
13 C) – 5
13 D) – 12
13
4) (15
,
20) Find csc θ.
A) 5
4B) 5
3C) 3
4D) 4
3
5) (7
,
9) Find tan θ.
A) 9
7B) 7
9C) 7
11 D) 9
11
6) (2
,
–3) Find sin θ.
A) – 313
13 B) 213
13 C) 13
2D) –3
7) (–3
,
–2) Find sec θ.
A) – 13
3B) 13
2C) 2
3D) – 313
13
Page 25
8) – 1
5, 1
2Find cos θ.
A) – 229
29 B) 529
29 C) – 29
5D) 29
2
9) – 1
2, 3
2Find cot θ.
A) – 3
3B) 23
3C) –3 D) –2
10) 2
2, – 2
2Find sec θ.
A) 2 B) –2 C) –1D)
– 2
2
11) (–5, –12) Find cot θ.
A) – 5
13 B) 5
12 C) 12
5D) – 12
13
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
12) (5, 12) Find sin θ.
2 Use Coterminal Angles to Find the Exact Value of a Trigonometric Function
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use a coterminal angle to find the exact value of the expression. Do not use a calculator.
1) cos –315°
A) 2
2B) 1
2C) – 1
2D) – 2
2
2) tan 750°
A) 3
3B) 3 C) –3 D) 3
2
3) sec 390°
A) 23
3B) 2 C) 3 D) 1
2
4) cos –660°
A) 1
2B) 2 C) 3
2D) 23
3
5) csc –360°
A) –1 B) 0 C) 1 D) undefined
Page 26
6) sin 19π
3
A) 3
2B) 1
2C) – 3
2D) – 1
2
7) csc 17π
4
A) 2 B) 23
3C) 2 D) 2
2
8) tan 25π
3
A) 3 B) 3
3C) 1 D) 3
2
9) tan (36π)
A) 0 B) 1 C) –1 D) undefined
3 Determine the Signs of the Trigonometric Functions of an Angle 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 C) III D) IV
2) cos θ
<
0, csc θ
<
0
A) I B) II C) III D) IV
3) sin θ > 0, cos θ
<
0
A) I B) II C) III D) IV
4) cot θ
<
0, cos θ > 0
A) I B) II C) III D) IV
5) csc θ > 0, sec θ > 0
A) I B) II C) III D) IV
6) sec θ
<
0, tan θ
<
0
A) I B) II C) III D) IV
7) tan θ
<
0, sin θ
<
0
A) I B) II C) III D) IV
8) cos θ > 0, csc θ
<
0
A) I B) II C) III D) IV
9) cot θ > 0, sin θ
<
0
A) I B) II C) III D) IV
10) sin θ > 0, cos θ > 0
A) I B) II C) III D) IV
Page 27
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) III only B) II and III C) II, III, and IV D) 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°)
4 Find the Reference Angle of an Angle
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the reference angle of the given angle.
1) 79°
A) 79° B) 11° C) 101° D) 169°
2) 103°
A) 77° B) 13° C) 87° D) 23°
3) 414°
A) 54° B) 36° C) 126° D) 144°
4) –373°
A) 13° B) 77° C) 167° D) 103°
5) –80°
A) 80° B) 10° C) 100° D) 170°
6) –244°
A) 64° B) 26° C) 154° D) 116°
7) –517°
A) 113° B) 23° C) 157° D) 67°
8) 13π
12
A) π
12 B) 11π
12 C) π
24 D) 13π
12
9) 3π
4
A) π
4B) 3π
4C) 5π
4D) π
8
Page 28
10) – 5π
4
A) π
4B) 3π
4C) 5π
4D) π
8
11) – 5π
6
A) π
6B) 5π
6C) 7π
6D) π
12
12) – 42π
8
A) π
4B) π
2C) π
3D) π
8
5 Use a Reference Angle to Find the Exact Value of a Trigonometric Function
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the reference angle to find the exact value of the expression. Do not use a calculator.
1) sin 855°
A) 2
2B) 1
2C) – 1
2D) – 2
2
2) tan 390°
A) 3
3B) 3 C) –3 D) 3
2
3) csc 960°
A) – 23
3B) –2 C) –3 D) – 1
2
4) cot 750°
A) 3 B) –3 C) 3
3D) – 3
3
5) sin 5π
3
A) – 3
2B) 3
2C) –1D)
– 1
2
6) tan –5π
6
A) 3
3B) 3 C) –3 D) 3
2
7) tan 5π
4
A) 1 B) 3 C) 3
3D) –1
Page 29
8) csc 4π
3
A) – 23
3B) –2 C) –3 D) – 1
2
9) sec –5π
4
A) –2 B) – 23
3C) –2D)
2
2
10) cot –5π
6
A) 3 B) –3 C) 3
3D) – 3
3
6 Find the Exact Values of Trigonometric Functions of an Angle, Given Information about the Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the indicated trigonometric function of θ.
1) cos θ = 4
9, tan θ < 0 Find sin θ.
A) – 65
9B) – 65
4C) – 9
4D) –65
2) sec θ = 3
2 , θ in quadrant IV Find tan θ.
A) – 5
2B) – 5
3C) – 3
2D) – 5
3) sin θ = – 2
3, tan θ > 0 Find sec θ.
A) – 35
5B) – 5
3C) 3
2D) – 25
5
4) csc θ = – 9
2, θ in quadrant III Find cot θ.
A) 77
2B) – 977
77 C) – 77
9D) – 277
77
5) tan θ = – 8
5, θ in quadrant II Find cos θ.
A) – 589
89 B) 589
89 C) 89
8D) – 89
5
6) cot θ = – 7
8, cos θ < 0 Find csc θ.
A) 113
8B) – 7 113
113 C) 7 113
113 D) – 113
7
Page 30
7) tan θ = – 7
24, 90°< θ < 180° Find cos θ.
A) – 24
25 B) –24 C) –24 31
31 D) –731
31
8) cos θ = 15
17, 3π
2 < θ < 2πFind cot θ.
A) – 15
8B) – 15 2
2C) – 8
15 D) 17
15
9) sin θ = 1
2, sec θ < 0 Find cos θ and tan θ.
A) cos θ = – 3
, tan θ = – 10 3
3B) cos θ = 3
2, tan θ = 3
3
C) cos θ = – 3
2, tan θ = – 3
3D) cos θ = – 3
2, tan θ = 3
3
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
10) sin θ = 1
6, sec θ < 0 Find cos θ and tan θ.
7 Demonstrate Additional Understanding and Skills
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) If sin θ = 0.6
,
find sin (θ + π).
A) –0.6 B) 0.6 C) 0.4 D) –0.4
2) If sin θ = 1
7, find csc θ.
A) 7 B) – 1
7C) 6
7D) undefined
3) Use a calculator to find the value of cos 118°. Round to the nearest hundredth.
A) –0.47 B) –1.88 C) 0.88 D) 0.19
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
4) Wildlife management personnel use predator–prey equations to model the populations of certain
predators and their prey in the wild. Suppose the population M of a predator after t months is given by
M = 750 + 125 sin π
6t
while the population N of its primary prey is given by
N = 12,250 + 3050 cos π
6t.
Calculate both M and N for t = 3, t = 15, and t = 27 months. Explain the answers in terms of reference
angles. Without calculating them, what will be the values of M and N for t = 39 months?
Page 31
5) The displacement d, in inches, from equilibrium of a weight suspended from a spring is given by
d = 1 + 2 sin(15t)°
where t in time in seconds. Find the displacement when t = 0, 2, 4, 6, 8, 10, and 12 seconds. Do not use a
calculator.
6) A study of ice cream consumption over 30 four–week periods in the early 1950s gives rise to the equation
C = 0.3520 + 0.0786 sin(0.4806θ – 0.1691)
where θ is the number (from 1 to 30) of the four–week period and C is the ice cream consumption in pints
per capita. Determine the consumption for the tenth four–week period. Round answer to the nearest 0.001
pint.
7.5 Unit Circle Approach; Properties of the Trigonometric Functions
1 Find the Exact Values of the Trigonometric Functions Using the Unit Circle
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
The point P on the unit circle that corresponds to a real number t is given. Find the indicated trigonometric
function.
1) 4
9, 65
9Find sin t.
A) 65
9B) 4
9C) 65
4D) 465
65
2) 3
8, 55
8Find tan t.
A) 55
3B) 8
3C) 55
8D) 355
55
3) 55
8, 3
8Find sec t.
A) 855
55 B) 8
3C) 55
3D) 355
55
4) – 77
9, 2
9Find cos t.
A) – 77
9B) 2
9C) – 77
2D) – 977
77
5) – 33
7, 4
7Find cot t.
A) – 33
4B) 4
7C) 33
7D) – 7
4
6) – 65
9, – 4
9Find sin t.
A) – 4
9B) – 65
9C) – 965
65 D) 9
4
Page 32
7) – 39
8, – 5
8Find cot t.
A) 39
5B) – 39
5C) – 539
39 D) 39
8
8) 2
9, – 77
9Find csc t.
A) – 977
77 B) – 77
9C) 77
2D) 77
9
9) 3
8, – 55
8Find cos t.
A) 3
8B) – 55
8C) 55
8D) – 3
8
10) 5
7, – 26
7Find csc t.
A) – 76
12 B) 7
5C) – 6
10 D) 5
7
The point P on the circle x2 + y2 = r2 that is also on the terminal side of an angle θ in standard position is given.
Find the indicated trigonometric function.
11) (–3
,
4) Find sin θ.
A) 4
5B) – 3
5C) – 4
5D) 3
5
12) (5
,
12) Find cos θ.
A) 5
13 B) 12
13 C) – 5
13 D) – 12
13
13) (–5
,
–4) Find tan θ.
A) 4
5B) 5
4C) 41
5D) – 41
5
14) (–5
,
4) Find cot θ.
A) – 5
4B) – 4
5C) 41
5D) – 41
5
15) (–2
,
–1) Find csc θ.
A) –5B) 5 C) –2D)
–5
16) (–3
,
–1) Find sec θ.
A) – 10
3B) 10
3C) – 310
10 D) –10
Page 33
2 Know the Domain and 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 from –1 to 1, inclusive
C) all real numbers, except odd multiples of π
2 (90°)
D) all real numbers, except integral multiples of π(180°)
2) For what numbers θ is f(θ) = tan θ not defined?
A) odd multiples of π
2 (90°) B) integral multiples of π (180°)
C) odd multiples of π (180°) D) all real numbers
3) For what numbers θ is f(θ) = cot θ not defined?
A) integral multiples of π (180°) B) odd multiples of π
2 (90°)
C) odd multiples of π (180°) D) all real numbers
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 1 or less than or equal to –1
D) all real numbers greater than or equal to 0
5) What is the range of the cotangent function?
A) all real numbers
B) all real numbers from –1 to 1, inclusive
C) all real numbers greater than or equal to 1 or less than or equal to –1
D) all real numbers, except integral multiples of π(180)°
6) What is the range of the secan
t
function?
A) all real numbers greater than or equal to 1 or less than or equal to –1
B) all real numbers from –1 to 1, inclusive
C) all real numbers
D) all real numbers, except odd multiples of π
2(90)°
3 Use the Periodic Properties to Find the Exact Values 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 855°
A) 2
2B) 1
2C) – 1
2D) – 2
2
Page 34
2) tan 570°
A) 3
3B) 3 C) –3 D) 3
2
3) csc 1020°
A) – 23
3B) –2 C) –3 D) – 1
2
4) cot 750°
A) 3 B) –3 C) 3
3D) – 3
3
5) cot 540°
A) 3 B) 0 C) 1 D) undefined
6) tan 720°
A) 0 B) 3
3C) –1 D) undefined
7) cos 20π
3
A) – 1
2B) – 3
2C) 3
2D) 1
2
8) sin 16π
3
A) – 3
2B) 3
2C) –1D)
– 1
2
9) tan 9π
4
A) 1 B) 3 C) 3
3D) –1
10) sec 11π
4
A) –2 B) – 23
3C) –2D)
2
2
Solve the problem.
11) If cos θ = 0.7
,
find the value of cos θ +cos (θ+2π) +cos (θ+4π).
A) 2.1 B) 2.1 +6πC) 4.1 D) 0.7
12) If cot θ = –7.1
,
find the value of cot θ+cot (θ+π) +cot (θ+2π).
A) –21.3 B) –21.3 +3πC) –19.3 D) undefined
Page 35
13) If f(θ) = sin θ and f(a) = – 1
3, find the exact value of f(a) + f(a + 2π) + f(a + 4π).
A) – 1B)
– 1 +6πC) 1 D) – 1
3
14) If f(θ) = tan θ and f(a) = 5
,
find the exact value of f(a) +f(a +π) +f(a +3π).
A) 15 B) 15 +4πC) 5 D) undefined
15) If f(x) = cos x and f(a) = – 1
12, find the exact value of f(a) + f(a – 2π) + f(a + 4π).
A) –12 B) –36 C) – 1
4D) – 1
12
16) If f(x) = sin x and f(a) = – 1
9, find the exact value of f(a) + f(a – 4π) + f(a – 2π).
A) – 1
3B) 1
3C) 1
6D) – 1
6
4 Use Even–Odd Properties to Find the Exact Values of the Trigonometric Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use the even–odd properties to find the exact value of the expression. Do not use a calculator.
1) sin (–30°)
A) – 1
2B) 1
2C) 3
2D) – 3
2
2) sin (–60°)
A) –3
2B) 1
2C) – 1
2D) 3
2
3) sec (–60°)
A) 2 B) –2C)
23
3D) –23
3
4) cot (–60°)
A) – 3
3B) 3
3C) 3 D) –3
5) cos (–150°)
A) – 3
2B) –1
2C) 1
2D) 3
2
6) cos – π
4
A) 2
2B) –2
2C) 3
2D) – 3
2
Page 36
7) csc – π
3
A) –23
3B) –2C)2 D)
23
3
8) cot – π
6
A) –3 B) 3
3C) 3 D) – 3
3
9) cot – π
2
A) 0 B) 1 C) –1 D) undefined
10) cos (–π)
A) –1 B) 1 C) 0 D) undefined
11) cot – π
4
A) –1B)
– 3
3C) –3 D) 1
Solve the problem.
12) If f(θ) = sin θ and f(a) = – 1
6, find the exact value of f(–a).
A) 1
6B) – 1
6C) 5
6D) – 5
6
13) If f(θ) = tan θ and f(a) = 5
,
find the exact value of f(–a).
A) –5B)5 C)
1
5D) – 1
5
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) Is the function f(θ) = sin θ + cos θ even, odd, or neither?
15) Is the function f(θ) = sin θ + tan θ even, odd, or neither?
5 Demonstrate Additional Understanding and Skills
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find the exact value of the expression. Do not use a calculator.
1) sin (–2π) + cos 5π
2
A) 0 B) 1 C) –2D)
–1
2) cos (2π) + sin –3π
2
A) 2 B) 0 C) –2D)1
Page 37
3) tan 9π
4 – cos 9π
4
A) 2 – 2
2B) 2 + 2
2C) –2 – 2
2D) –2 + 2
2
4) csc 5π
2 – sec –8π
3
A) 3 B) –1C)1 D)
–3
Page 38
7.6 Graphs of the Sine and Cosine Functions
1 Graph Functions of the Form y = A sin(ωx) Using Transformations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use transformations to graph the function.
1) y = 5 sin x
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 39
2) y = sin (x + π)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 40
3) y = sin x – 4
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 41
4) y = –5 sin x
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 42
5) y = sin (πx)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 43
6) y = 5 sin x – 1
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 44
7) y = –2 sin (x + π
2)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 45
8) y = 4 sin (π – x)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Solve the problem.
9) For what numbers x, 0 ≤ x ≤ 2π
,
does sin x =0?
A) 0, π
,
2πB) π
2, 3π
2C) 0, 1 D) 0, 1, 2
10) For what numbers x, 0 ≤ x ≤ 2π
,
does sin x =1?
A) π
2B) π
2, 3π
2C) 0, 2πD) none
Page 46
11) For what numbers x, 0 ≤ x ≤ 2π
,
does sin x = –1?
A) 3π
2B) π
2, 3π
2C) πD) none
2 Graph Functions of the Form y = A cos(ωx) Using Transformations
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Use transformations to graph the function.
1) y = 4 cos x
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 47
2) y = cos (x + π
3)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 48
3) y = cos x + 5
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 49
4) y = –4 cos x
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 50
5) y = cos (π
4x)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 51
6) y = 2 cos x + 4
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
–
23
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 52
7) y = –3 cos (x – π
4)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 53
8) y = 5 cos (π – x)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Solve the problem.
9) What is the y–intercept of y = sin x?
A) 0 B) 1 C) πD) π
2
10) For what numbers x, 0 ≤ x ≤ 2π
,
does cos x =0?
A) π
2, 3π
2B) 0, π
,
2πC) 0, 1 D) 0, 1, 2
Page 54
11) For what numbers x, 0 ≤ x ≤ 2π
,
does cos x =1?
A) 0, 2πB) π
2, 3π
2C) π
2D) none
12) For what numbers x, 0 ≤ x ≤ 2π
,
does cos x = –1?
A) πB) π
2, 3π
2C) π
2D) none
3 Determine the Amplitude and Period of Sinusoidal Functions
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Without graphing the function, determine its amplitude or period as requested.
1) y = –2 sin x Find the amplitude.
A) 2 B) –2πC) π
2D) 2π
2) y = 3 sin 1
2x Find the amplitude.
A) 3 B) 3π
2C) π
3D) 4π
3) y = –5 sin 4x Find the amplitude.
A) 5 B) 5
4C) π
5D) π
4
4) y = sin 5x Find the period.
A) 2π
5B) 5 C) 2πD) 1
5) y = –3 cos 1
2x Find the amplitude.
A) 3 B) 3π
2C) π
3D) 4π
6) y = cos 5x Find the period.
A) 2π
5B) 5 C) 2πD) 1
7) y = –3 cos 1
4x Find the period.
A) 8πB) –3C)
3π
4D) π
4
8) y = –3 cos x Find the period.
A) 2πB) 3 C) π
3D) π
Page 55
9) y = 5
6 sin (– 6π
7x) Find the period.
A) 7
3B) 12π
7C) 5π
3D) 3
5
10) y = 3
8 cos (– 6π
7x) Find the amplitude.
A) 3
8B) 7
3C) 8π
3D) 6π
7
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
Solve the problem.
11) Wildlife management personnel use predator–prey equations to model the populations of certain
predators and their prey in the wild. Suppose the population M of a predator after t months is given by
M = 750 + 125 sin π
6t
while the population N of its primary prey is given by
N = 12,250 + 3050 cos π
6t
Find the period for each of these functions.
12) The average daily temperature T of a city in the United States is approximated by
T = 55 – 23 cos 2π
365 (t –30)
where t is in days, 1 ≤ t ≤ 365, and t = 1 corresponds to January 1. Find the period of T.
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
13) The current I, in amperes, flowing through a particular ac (alternating current) circuit at time t seconds is
I = 110 sin (50πt)
What is the period and amplitude of the current?
A) period = 1
25 second, amplitude = 110 B) period = 50π seconds, amplitude = 1
25
C) period = 1
300 second, amplitude = 300 D) period = π
110 second, amplitude = 50
Page 56
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
14) The current I, in amperes, flowing through an ac (alternating current) circuit at time t, in seconds, is
I = 30 sin(50πt)
What is the amplitude? What is the period?
Graph this function over two periods beginning at t = 0.
t
I
t
I
15) A mass hangs from a spring which oscillates up and down. The position P of the mass at time t is given by
P = 4 cos(4t)
What is the amplitude? What is the period?
Graph this function over two periods beginning at t = 0.
t
2
P
4
-4
t
2
P
4
-4
Page 57
16) Before exercising, an athlete measures her air flow and obtains
a = 0.65 sin 2π
5t
where a is measured in liters per second and t is the time in seconds. If a > 0, the athlete is inhaling; if a < 0,
the athlete is exhaling. The time to complete one complete inhalation/exhalation sequence is a respiratory
cycle.
What is the amplitude? What is the period? What is the respiratory cycle?
Graph a over two periods beginning at t = 0.
t
510
a
1
-1
t
510
a
1
-1
17) A boy is flying a model airplane while standing on a straight line. The plane, at the end of a twenty–five
foot wire, flies in circles around the boy. The directed distance of the plane from the straight line is found
to be
d = 25 cos 3π
4t
where d is measured in feet and t is the time in seconds. If d > 0, the plane is in front of the boy; if d <0, th
e
plane is behind him.
What is the amplitude? What is the period?
Graph d over two periods beginning at t = 0.
t
4
3
8
3416
3
d
25
-25
t
4
3
8
3416
3
d
25
-25
Page 58
4 Graph Sinusoidal Functions Using Key Points
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Match the given function to its graph.
1) 1) y = sin x 2) y = cos x
3) y = –sin x 4) y = –cos x
AB
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
CD
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A) 1C, 2A, 3B, 4D B) 1A, 2D, 3C, 4B C) 1B, 2D, 3C, 4A D) 1A, 2B, 3C, 4D
Page 59
2) 1) y = sin(3x) 2) y = 3 cos x
3) y = 3 sin x 4) y = cos(3x)
AB
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
CD
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A) 1B, 2D, 3C, 4A B) 1A, 2D, 3C, 4B C) 1A, 2C, 3D, 4B D) 1A, 2B, 3C, 4D
Page 60
3) 1) y = sin x – π
22) y = cos x + π
2
3) y = sin x + π
24) y = cos x – π
2
AB
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
CD
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A) 1C, 2A, 3B, 4D B) 1A, 2D, 3C, 4B C) 1B, 2D, 3C, 4A D) 1A, 2B, 3C, 4D
Page 61
4) 1) y = 1 + sin x 2) y = 1 + cos x
3) y = –1 + sin x 4) y = –1 + cos x
AB
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
CD
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A) 1B, 2D, 3C, 4A B) 1A, 2D, 3C, 4B C) 1A, 2C, 3D, 4B D) 1A, 2B, 3C, 4D
Page 62
5) 1) y = sin 1
4x 2) y = 1
4 cos x
3) y = 1
4 sin x 4) y = cos 1
4x
AB
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
CD
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
A) 1B, 2D, 3C, 4A B) 1A, 2D, 3C, 4B C) 1A, 2C, 3D, 4B D) 1A, 2B, 3C, 4D
Page 63
6) 1) y = –3 sin(2x) 2) y = –3 sin 1
2x
3) y = –3 cos(2x) 4) y = –3 cos 1
2x
AB
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
CD
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
A) 1A, 2C, 3D, 4B B) 1C, 2A, 3B, 4D C) 1C, 2A, 3D, 4B D) 1D, 2B, 3A, 4C
Page 64
7) 1) y = –3 sin π
2x 2) y = –3 sin 1
2x
3) y = –3 cos π
2x 4) y = –3 cos 1
2x
A) B)
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
C) D)
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
A) 1B, 2D, 3A, 4C B) 1A, 2C, 3B, 4D C) 1C, 2A, 3D, 4B D) 1A, 2C, 3D, 4B
Answer the question.
8) Which one of the equations below matches the graph?
A) y = 3 cos 1
4x B) y = 3 cos(4x) C) y = 3 sin 1
4x D) y = –3 sin(4x)
Page 65
9) Which one of the equations below matches the graph?
A) y = 4 cos(2x) B) y = 4 sin 1
2x C) y = 2 cos 1
4x D) y = 4 cos 1
2x
10) Which one of the equations below matches the graph?
A) y = 2 sin 1
3x B) y = –2 sin 1
3x C) y = 2 cos 1
3x D) y =2 cos(3x)
11) Which one of the equations below matches the graph?
A) y = –2 sin(3x) B) y = –2 sin 1
3x C) y = 2 sin 1
3x D) y = –2 cos(3x)
Page 66
Graph the sinusoidal function using key points.
12) y = 4 sin(πx)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 67
13) y = –3 cos(πx)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 68
14) y = 2 sin(3x)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 69
15) y = –3 sin 1
4x
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
A)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
B)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
C)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
D)
x
–
23
y
6
4
2
-2
-4
-6
x
–
23
y
6
4
2
-2
-4
-6
Page 70
16) y = 2 + sin x
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
B)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
C)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
D)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
Page 71
17) y = 2 + cos x
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
B)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
C)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
D)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
Page 72
18) y = sin x – 2
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
B)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
C)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
D)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
Page 73
19) y = cos x – 2
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
A)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
B)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
C)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
D)
x
-2–
2
y
3
2
1
-1
-2
-3
x
-2–
2
y
3
2
1
-1
-2
-3
Page 74
20) y = –4 sin 1
2x – 2
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
A)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
B)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
C)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
D)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
Page 75
21) y = 4 cos 1
2x – 2
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
A)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
B)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
C)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
D)
x
-2–
2
y
6
4
2
-2
-4
-6
x
-2–
2
y
6
4
2
-2
-4
-6
Page 76
22) y = 4
3 cos – 1
2x
x
-22468
y
6
4
2
-2
-4
-6
x
-22468
y
6
4
2
-2
-4
-6
A)
x
-22468
y
6
4
2
-2
-4
-6
x
-22468
y
6
4
2
-2
-4
-6
B)
x
-22468
y
6
4
2
-2
-4
-6
x
-22468
y
6
4
2
-2
-4
-6
C)
x
-22468
y
6
4
2
-2
-4
-6
x
-22468
y
6
4
2
-2
-4
-6
D)
x
-22468
y
6
4
2
-2
-4
-6
x
-22468
y
6
4
2
-2
-4
-6
Page 77
5 Find an Equation for a Sinusoidal Graph
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Write the equation of a sine function that has the given characteristics.
1) Amplitude: 3
Period: 6π
A) y = 3 sin 1
3x B) y = 3 sin (6x) C) y =sin (6x) +3D)y
= 6 sin 2
3x
2) Amplitude: 4
Period: 6
A) y = 4 sin 1
3πx B) y = 4 sin (6x) C) y =sin (6πx) +4D)y
= 6 sin 1
2πx
Find an equation for the graph.
3)
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = 3 sin 1
2x B) y = 3 sin (2x) C) y = 2 sin 1
3x D) y =2 sin (3x)
4)
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = 5 cos 1
2x B) y = 5 cos (2x) C) y = 2 cos 1
5x D) y =2 cos (5x)
Page 78
5)
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = –4 sin 1
3x B) y = –4 sin (3x) C) y = –4 cos 1
3x D) y = –4 cos (3x)
6)
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = 5 sin (3x) B) y = 5 sin 1
3x C) y = 3 sin 1
5x D) y =3 sin (5x)
7)
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = 3 cos (2x) B) y = 3 cos 1
2x C) y = 2 cos 1
3x D) y =2 cos (3x)
Page 79
8)
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-2–
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = –5 cos (2x) B) y = –5 cos 1
2x C) y = –5 sin 1
2x D) y = –5 sin (2x)
9)
x
-1
2
1
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-1
2
1
2
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = 3 sin (2πx) B) y = 3 sin π
2x C) y =2 sin (3πx) D) y = 2 sin π
3x
10)
x
-5 -4 -3 -2 -1 1 2 3 4 5 6
y
5
4
3
2
1
-1
-2
-3
-4
-5
x
-5 -4 -3 -2 -1 1 2 3 4 5 6
y
5
4
3
2
1
-1
-2
-3
-4
-5
A) y = 4 cos π
3x B) y = 4 cos (3πx) C) y =3 cos (4πx) D) y = 3 cos π
4x
Page 80
11)
A) y = 4 cos (2x) B) y = 4 cos 1
2x C) y =4 sin (2x) D) y = –4 cos (2x)
12)
A) y = –5 sin 1
3x B) y = 5 cos 1
3x C) y = –5 sin (3x) D) y = –5 sin 2
3x
13)
A) y = –3 cos 1
3x B) y = 3 cos 1
3x C) y = –3 sin (3x) D) y = –3 cos (3x)
Page 81
14)
A) y = 1
2 cos (4x) B) y = 1
2 cos 1
2x C) y = 1
2 cos 1
4x D) y =cos (4x)
7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions
1 Graph Functions of the Form y = A tan (ωx) +B and y =A cot (ωx) +B
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) What is the y–intercept of y = cot x?
A) π
2B) 0 C) 1 D) none
2) For what numbers x, –2π ≤ x ≤ 2π
,
does the graph of y =tan x have vertical asymptotes.?
A) – 3π
2, – π
2, π
2, 3π
2B) –2π
,
–π
,
0, π
,
2πC) –2, –1, 0, 1, 2 D) none
Graph the function.
3) y = 2 tan x
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
Page 82
A)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
B)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
C)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
D)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
4) y = 3 tan(4x)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
Page 83
A)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
B)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
C)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
D)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
5) y = –4 tan 1
2x
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
Page 84
A)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
B)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
C)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
D)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
6) y =cot(πx)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
Page 85
A)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
B)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
C)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
D)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
7) y =cot(4x)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
Page 86
A)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
B)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
C)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
D)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
8) y = –4 cot(4x)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
Page 87
A)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
B)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
C)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
D)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
9) y = –cot x
x
––
2
23
225
23
y
3
-3
x
––
2
23
225
23
y
3
-3
Page 88
A)
x
––
2
23
225
23
y
3
-3
x
––
2
23
225
23
y
3
-3
B)
x
––
2
23
225
23
y
3
-3
x
––
2
23
225
23
y
3
-3
C)
x
––
2
23
225
23
y
3
-3
x
––
2
23
225
23
y
3
-3
D)
x
––
2
23
225
23
y
3
-3
x
––
2
23
225
23
y
3
-3
10) y = 2
3 cot x
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
Page 89
A)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
B)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
C)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
D)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
11) y = cot x + 2
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
Page 90
A)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
B)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
C)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
D)
x
––
2
2
y
4
2
-2
-4
x
––
2
2
y
4
2
-2
-4
Solve the problem.
12) Find the average rate of change of f(x) = tan x from 0 to π
4.
A) 4
πB) π
4C) 0 D) 1
2 Graph Functions of the Form y = A csc (ωx) +B and y =A sec (ωx) +B
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) What is the y–intercept of y = sec x?
A) 1 B) 0 C) π
2D) none
2) For what numbers x, –2π ≤ x ≤ 2π
,
does the graph of y =csc x have vertical asymptotes?
A) –2π
,
–π
,
0, π
,
2πB) – 3π
2, – π
2, π
2, 3π
2C) –2, –1, 0, 1, 2 D) none
Page 91
Graph the function.
3) y = –csc x
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
A)
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
B)
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
C)
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
D)
x
-2–
2
y
3
-3
x
-2–
2
y
3
-3
Page 92
4) y = sec(2x)
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-2–
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 93
5) y = csc 1
2x
x
-22468
y
3
-3
x
-22468
y
3
-3
A)
x
-22468
y
3
-3
x
-22468
y
3
-3
B)
x
-22468
y
3
-3
x
-22468
y
3
-3
C)
x
-22468
y
3
-3
x
-22468
y
3
-3
D)
x
-22468
y
3
-3
x
-22468
y
3
-3
Page 94
6) y = 2 sec 1
4x
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
-22468
y
10
8
6
4
2
-2
-4
-6
-8
-10
Page 95
7) y = 3 csc(2x)
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
A)
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
B)
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
C)
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
D)
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
x
––
2
2
y
10
8
6
4
2
-2
-4
-6
-8
-10
Solve the problem.
8) Find the average rate of change of f(x) = sec x from 0 to π
4.
A) 4( 2 – 1)
πB) 42
– 1
πC) 2 – 1
4πD) 2 – 1
π
Page 96
9) A rotating beacon is located 5 ft from a wall. If the distance from the beacon to the point on the wall
where the beacon is aimed is given by a = 5|sec (2πt)| , where t is in seconds, find a when t = 0.44 seconds.
Round your answer to the nearest hundredth.
A) 5.38 ft B) 26.68 ft C) 16.03 ft D) –5.38 ft
7.8 Phase Shift; Sinusoidal Curve Fitting
1 Graph Sinusoidal Functions of the Form y =A sin (ωx –φ) +B
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Find (i) the amplitude, (ii) the period, and (iii) the phase shift.
1) y = – 1
2 sin(4x + 3π)
A) (i) 1
2(ii) π
2(iii) – 3π
4B) (i) 2 (ii) π
2(iii) 3π
C) (i) – 1
2(ii) 4 (iii) – 4π
3D) (i) 1
2(ii) 4 (iii) – 3π
4
2) y = – 1
2 cos(2x – 2π)
A) (i) 1
2(ii) π(iii) πB) (i) 1
2(ii) π(iii) π
2
C) (i) 2 (ii) 2π(iii) 2πD) (i) 2 (ii) π(iii) π
Find the amplitude.
3) y = –3 cos 2x + π
4
A) 3 B) 2 C) π
2D) –6
4) y = 4 sin 3x + π
3
A) 4 B) 3 C) π
3D) 12
5) y = –3 cos(4x – π)
A) 3 B) 4 C) πD) –12
6) y = –2 sin(3x + π)
A) 2 B) 3 C) πD) –6
7) y = –4 cos(3πx – 3)
A) 4 B) 3πC) πD) –12
8) y = –4 sin(2πx + 3)
A) 4 B) 2πC) πD) –8
Page 97
Find the period.
9) y = 3 sin 8x + π
2
A) π
4B) 3 C) 8 D) π
10) y = –3 cos 5x + π
2
A) 2π
5B) 3 C) π
2D) π
11) y = 5 sin 1
4x – π
2
A) 8πB) 5πC) 4πD) π
2
12) y = 5 cos 1
2x + π
3
A) 4πB) 5πC) 2πD) 2π
3
13) y = 5 sin(6πx + 4)
A) 1
3B) π
3C) 6πD) 1
6
14) y = 3 cos(3πx + 4)
A) 2
3B) 2π
3C) 3πD) 1
3
Find the phase shift.
15) y = –5 sin x – π
2
A) π
2 units to the right B) π
2 units to the left
C) –5 units up D) –5 units down
16) y = –5 cos x + π
2
A) π
2 units to the left B) π
2 units to the right
C) –5 units up D) –5 units down
Page 98
17) y = –5 sin 4x – π
2
A) π
8 units to the right B) π
2 units to the left
C) 5π units up D) 4πunits down
18) y = –5 cos(4x + π)
A) π
4 units to the left B) π
5 units to the left
C) 5π units to the right D) 4πunits to the right
19) y = 3 sin 1
4x – π
4
A) π units to the right B) π
4 units to the right
C) π
16 units to the left D) π
3 units to the left
20) y = –2 cos 1
2x + π
2
A) π units to the left B) π
2 units to the left
C) π
4 units to the right D) 2πunits to the right
21) y = 4 sin(2πx – 3)
A) 3
2π units to the right B) 3
2 units to the left
C) 3 units to the right D) 3 units to the left
22) y = 3 cos –3x + π
3
A) π
9 units to the right B) π
9 units to the left
C) π
3 units to the right D) π
3 units to the left
Write the equation of a sine function that has the given characteristics.
23) Amplitude: 4
Period: 5π
Phase Shift: π
5
A) y = 4 sin 2
5x – 2
25πB) y = 4 sin 5
2x + 2
25π
C) y = 4 sin 2
5x + 2
25πD) y = 4 sin 5x + π
5
Page 99
24) Amplitude: 5
Period: 4π
Phase Shift: – π
4
A) y = 5 sin 1
2x + 1
8πB) y = 5 sin 2x – 1
8πC) y = 5 sin 1
2x – 1
8πD) y = 5 sin 4x – π
4
25) Amplitude: 4
Period: π
Phase Shift: – 5
A) y = 4 sin(2x + 10) B) y = 4 sin 1
2x – 10 C) y =4 sin(x –5) D) y =sin(4x +5)
26) Amplitude: 2
Period: π
Phase Shift: 5
2
A) y = 2 sin(2x – 5) B) y = 2 sin 1
2x – 10 C) y = 2 sin 2x + 5
2D) y =sin(2x +5)
Graph the function. Show at least one period.
27) y = 4 sin(2πx + 2)
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
A)
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
B)
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
Page 100
C)
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
D)
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
x
-1 1
y
4
3
2
1
-1
-2
-3
-4
28) y = 3 sin(5x – π)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
A)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
B)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
Page 101
C)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
D)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
29) y = 4 sin(–4x – π)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
A)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
B)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
Page 102
C)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
D)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
30) y = 2 cos 5x + π
2
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
A)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
B)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
Page 103
C)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
D)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
31) y = –5 sin 4x + π
2
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
A)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
B)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
Page 104
C)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
D)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
32) y = 4 sin(πx + 5)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
A)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
B)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
Page 105
C)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
D)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
33) y = 3 cos –2x + π
2
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
A)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
B)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
Page 106
C)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
D)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
34) y = –4 cos(πx + 5)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
A)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
B)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
Page 107
C)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
D)
x
–
23
y
8
6
4
2
-2
-4
-6
-8
x
–
23
y
8
6
4
2
-2
-4
-6
-8
35) y = 3 sec 2x – π
5
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
A)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
B)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
Page 108
C)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
D)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
36) y = 3 csc 2x – π
6
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
A)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
B)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
Page 109
C)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
D)
x
––
2
2
y
6
4
2
-2
-4
-6
x
––
2
2
y
6
4
2
-2
-4
-6
37) y = –2 tan x + π
4
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
A)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
B)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
Page 110
C)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
D)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
38) y = 1
2 cot x – π
4
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
A)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
B)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
Page 111
C)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
D)
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
x
––
2
23
225
23
y
6
4
2
-2
-4
-6
2 Find a Sinusoidal Function from Data
MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Solve the problem.
1) An experiment in a wind tunnel generates cyclic waves. The following data is collected for 60 seconds:
Time
(in seconds)
Wind speed
(in feet per second)
025
15 50
30 75
45 50
60 25
Let V represent the wind speed (velocity) in feet per second and let t represent the time in seconds. Write a
sine equation that describes the wave.
A) V = 25 sin π
30t – π
2 + 50 B) V =75 sin(60t –30) + 25
C) V = 75 sin π
30t – π
2 + 25 D) V =50 sin (60t –30) + 25
Page 112
2) A town’s average monthly temperature data is represented in the table below:
Month, x
Average Monthly
Temperature, °F
January, 1
February, 2
March, 3
April, 4
May, 5
June, 6
July, 7
August, 8
September, 9
October, 10
November, 11
December, 12
28.9
31.9
44.0
56.7
70.6
78.9
83.6
78.9
80.9
57.3
41.2
32.2
Find a sinusoidal function of the form y = A sin (ωx – φ) + B that fits the data.
A) y = 27.35 sin π
6x – 2π
3 + 56.25 B) y = 83.6 sin π
6x – 2π
3 + 28.9
C) y = 56.25 sin π
6x – π
4 + 27.35 D) y = 28.9 sin π
6x – π
4 + 83.6
3) The number of hours of sunlight in a day can be modeled by a sinusoidal function. In the northern
hemisphere, the longest day of the year occurs at the summer solstice and the shortest day occurs at the
winter solstice. In 2000, these dates were June 22 (the 172nd day of the year) and December 21 (the 356th
day of the year), respectively.
A town experiences 11.29 hours of sunlight at the summer solstice and 7.94 hours of sunlight at the winter
solstice. Find a sinusoidal function y = A sin (ωx – φ) + B that fits the data, where x is the day of the year.
(Note: There are 366 days in the year 2000.)
A) y = 1.675 sin π
183 x – 161π
366 + 9.615 B) y = 11.29 sin πx – 2π
3 + 7.94
C) y = 1.675 sin π
183 x – 2π
3 + 9.615 D) y = 11.29 sin 172π
356 x – 2π
3 + 9.615
Page 113
4) The data below represent the average monthly cost of natural gas in an Oregon home.
Month Aug Sep Oct Nov Dec Jan
Cost 21.20 28.24 44.73 67.25 89.77 106.26
Month Feb Mar Apr May Jun Jul
Cost 111.30 106.26 89.77 67.25 43.73 28.24
Above is the graph of 45.05 sin x superimposed over a scatter diagram of the data. Find the sinusoidal
function of the form y = A sin (ωx – φ) + B which best fits the data.
A) y = 45.05 sin π
6x – 2π
3 + 66.25 B) y = 45.05 sin π
8t + 12 + 21.20
C) y = 45.05 sin π
4x – 2π
3 + 21.20 D) y = 45.05 sin π
6x – π
12 + 66.25
Page 114
SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.
5) The data below represent the average monthly cost of natural gas in an Oregon home.
Month Aug Sep Oct Nov Dec Jan
Cost 18.90 24.24 44.58 68.25 91.92 109.26
Month Feb Mar Apr May Jun Jul
Cost 113.60 106.26 91.92 68.25 42.58 24.24
Above is the graph of 47.35 sin x. Make a scatter diagram of the data. Find the sinusoidal function of the
form y = A sin (ωx – φ) + B which fits the data.
Page 115
6) The following data represents the normal monthly precipitation for a certain city in California.
Month, x
Normal Monthly
Precipitation, inches
January, 1
February, 2
March, 3
April, 4
May, 5
June, 6
July, 7
August, 8
September, 9
October, 10
November, 11
December, 12
6.06
4.45
4.38
2.08
1.27
0.56
0.17
0.46
0.91
2.24
5.21
5.51
Draw a scatter diagram of the data for one period. Find a sinusoidal function of the form
y = A sin (ωx – φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a
graphing utility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the
scatter diagram.
x
y
x
y
Page 116
7) The following data represents the normal monthly precipitation for a certain city in Arkansas.
Month, x
Normal Monthly
Precipitation, inches
January, 1
February, 2
March, 3
April, 4
May, 5
June, 6
July, 7
August, 8
September, 9
October, 10
November, 11
December, 12
3.91
4.36
5.31
6.21
7.02
7.84
8.19
8.06
7.41
6.30
5.21
4.28
Draw a scatter diagram of the data for one period. Find the sinusoidal function of the form
y = A sin (ωx – φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a
graphing utility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the
scatter diagram.
x
y
x
y
Page 117
8) The following data represents the average monthly minimum temperature for a certain city in California.
Month, x
Average Monthly
Minimum
Temperature, °F
January, 1
February, 2
March, 3
April, 4
May, 5
June, 6
July, 7
August, 8
September, 9
October, 10
November, 11
December, 12
49.6
50.8
55.6
57.5
60.7
63.6
65.9
65.6
64.4
62.1
54.2
50.1
Draw a scatter diagram of the data for one period. Find a sinusoidal function of the form
y = A sin (ωx – φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a
graphing utility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the
scatter diagram.
x
y
x
y
Page 118
9) The following data represents the average percent of possible sunshine for a certain city in Indiana.
Month, x
Average Percent of
Possible Sunshine
January, 1
February, 2
March, 3
April, 4
May, 5
June, 6
July, 7
August, 8
September, 9
October, 10
November, 11
December, 12
46
51
55
60
68
73
75
74
68
62
41
38
Draw a scatter diagram of the data for one period. Find the sinusoidal function of the form
y = A sin (ωx – φ) + B that fits the data. Draw the sinusoidal function on the scatter diagram. Use a
graphing utility to find the sinusoidal function of best fit. Draw the sinusoidal function of best fit on the
scatter diagram.
x
y
x
y
Page 119
Ch. 7 Trigonometric Functions
Answer Key
7.1 Angles and Their Measure
1 Convert between Decimals and Degrees, Minutes, Seconds Measures for Angles
Page 120
4 Find the Area of a Sector of a Circle
5 Find the Linear Speed of an Object Traveling in Circular Motion
7.2 Right Triangle Trigonometry
1 Find the Values of Trigonometric Functions of Acute Angles
Page 121
2 Use the Fundamental Identities
3 Find the Values of the Remaining Trigonometric Functions, Given the Value of One of Them
4 Use the Complementary Angle Theorem
7.3 Computing the Values of Trigonometric Functions of Acute Angles
1 Find the Exact Values of the Trigonometric Functions of π/4 =45°
Page 122
2 Find the Exact Values of the Trigonometric Functions of π/6 =30° and π/3 =60°
3 Use a Calculator to Approximate the Values of the Trigonometric Functions of Acute Angles
4 Model and Solve Applied Problems Involving Right Triangles
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7.4 Trigonometric Functions of Any Angle
1 Find the Exact Values of the Trigonometric Functions for Any Angle
2 Use Coterminal Angles to Find the Exact Value of a Trigonometric Function
4 Find the Reference Angle of an Angle
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6 Find the Exact Values of Trigonometric Functions of an Angle, Given Information about the Functions
7 Demonstrate Additional Understanding and Skills
7.5 Unit Circle Approach; Properties of the Trigonometric Functions
1 Find the Exact Values of the Trigonometric Functions Using the Unit Circle
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2 Know the Domain and Range of the Trigonometric Functions
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7.6 Graphs of the Sine and Cosine Functions
2 Graph Functions of the Form y = A cos(ωx) Using Transformations
3 Determine the Amplitude and Period of Sinusoidal Functions
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4 Graph Sinusoidal Functions Using Key Points
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7.7 Graphs of the Tangent, Cotangent, Cosecant, and Secant Functions
1 Graph Functions of the Form y = A tan (ωx) +B and y =A cot (ωx) +B
2 Graph Functions of the Form y = A csc (ωx) +B and y =A sec (ωx) +B
7.8 Phase Shift; Sinusoidal Curve Fitting
1 Graph Sinusoidal Functions of the Form y =A sin (ωx –φ) +B
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2 Find a Sinusoidal Function from Data
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