Precalculus Student Solutions Manual 5th

Margaret L. Lial, John Hornsby, David I. Schneider

Chapter 5

Trigonometric Functions - all with Video Answers

Educators

Section 1

Angles

Problem 1

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$30^{\circ}$$

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Problem 2

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$60^{\circ}$$

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Problem 3

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$45^{\circ}$$

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Problem 4

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$18^{\circ}$$

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Problem 5

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$54^{\circ}$$

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Problem 6

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$89^{\circ}$$

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Problem 7

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$1^{\circ}$$

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Problem 8

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$10^{\circ}$$

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Problem 9

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$14^{\circ} 20^{\prime}$$

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Problem 10

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$39^{\circ} 50^{\prime}$$

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Problem 11

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$20^{\circ} 10^{\prime} 30^{\prime \prime}$$

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Problem 12

Find (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3
$$50^{\circ} 40^{\prime} 50^{\prime \prime}$$

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Problem 13

Find the measure of each unknown angle in Exercises $13-22 .$ See Example $2 .$
(FIGURE CAN NOT COPY)

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Problem 14

Find the measure of each unknown angle in Exercises $13-22 .$ See Example $2 .$
(FIGURE CAN NOT COPY)

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Problem 15

Find the measure of each unknown angle in Exercises $13-22 .$ See Example $2 .$
(FIGURE CAN NOT COPY)

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Problem 16

Find the measure of each unknown angle in Exercises $13-22 .$ See Example $2 .$
(FIGURE CAN NOT COPY)

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Problem 17

Find the measure of each unknown angle in Exercises $13-22 .$ See Example $2 .$
(FIGURE CAN NOT COPY)

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Problem 18

Find the measure of each unknown angle in Exercises $13-22 .$ See Example $2 .$
(FIGURE CAN NOT COPY)

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Problem 19

Supplementary angles with measures $10 x+7$ and $7 x+3$ degrees

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Problem 20

Supplementary angles with measures $6 x-4$ and $8 x-12$ degrees

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Problem 21

Complementary angles with measures $9 x+6$ and $3 x$ degrees

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Problem 22

complementary angles with measures $3 x-5$ and $6 x-40$ degrees

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Problem 23

What is the measure of an angle that is its own complement?

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Problem 24

What is the measure of an angle that is its own supplement?

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Problem 25

Find the measure of the smaller angle formed by the hands of a clock at the following times.
(FIGURE CAN NOT COPY)

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Problem 26

Find the measure of the smaller angle formed by the hands of a clock at the following times.
(FIGURE CAN NOT COPY)

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Problem 27

Find the measure of the smaller angle formed by the hands of a clock at the following times.
$$3: 15$$

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Problem 28

Find the measure of the smaller angle formed by the hands of a clock at the following times.
$$9: 45$$

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Problem 29

Find the measure of the smaller angle formed by the hands of a clock at the following times.
$$8: 20$$

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Problem 30

Find the measure of the smaller angle formed by the hands of a clock at the following times.
$$6: 10$$

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Problem 31

If an angle measures $x^{\circ},$ how can we represent its complement?

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Problem 32

If an angle measures $x^{\circ},$ how can we represent its supplement?

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Problem 33

If a positive angle has measure $x^{\circ}$ between $0^{\circ}$ and $60^{\circ},$ how can we represent the first negative angle coterminal with it?

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Problem 34

If a negative angle has measure $x^{\circ}$ between $0^{\circ}$ and $-60^{\circ},$ how can we represent the first positive angle coterminal with it?

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Problem 35

Perform each calculation. See Example 3 .
$$62^{\circ} 18^{\prime}+21^{\circ} 41^{\prime}$$

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Problem 36

Perform each calculation. See Example 3 .
$$75^{\circ} 15^{\prime}+83^{\circ} 32^{\prime}$$

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Problem 37

Perform each calculation. See Example 3 .
$$97^{\circ} 42^{\prime}+81^{\circ} 37^{\prime}$$

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Problem 38

Perform each calculation. See Example 3 .
$$110^{\circ} 25^{\prime}+32^{\circ} 55^{\prime}$$

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Problem 39

Perform each calculation. See Example 3 .
$$71^{\circ} 18^{\prime}-47^{\circ} 29^{\prime}$$

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Problem 40

Perform each calculation. See Example 3 .
$$47^{\circ} 23^{\prime}-73^{\circ} 48^{\prime}$$

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Problem 41

Perform each calculation. See Example 3 .
$$90^{\circ}-51^{\circ} 28^{\prime}$$

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Problem 42

Perform each calculation. See Example 3 .
$$90^{\circ}-17^{\circ} 13^{\prime}$$

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Problem 43

Perform each calculation. See Example 3 .
$$180^{\circ}-119^{\circ} 26^{\prime}$$

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Problem 44

Perform each calculation. See Example 3 .
$$180^{\circ}-124^{\circ} 51^{\prime}$$

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Problem 45

Perform each calculation. See Example 3 .
$$26^{\circ} 20^{\prime}+18^{\circ} 17^{\prime}-14^{\circ} 10^{\prime}$$

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Problem 46

Perform each calculation. See Example 3 .
$$55^{\circ} 30^{\prime}+12^{\circ} 44^{\prime}-8^{\circ} 15^{\prime}$$

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Problem 47

Perform each calculation. See Example 3 .
$$90^{\circ}-72^{\circ} 58^{\prime} 11^{\prime \prime}$$

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Problem 48

Perform each calculation. See Example 3 .
$$90^{\circ}-36^{\circ} 18^{\prime} 47^{\prime \prime}$$

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Problem 49

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$35^{\circ} 30^{\prime}$$

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Problem 50

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$82^{\circ} 30^{\prime}$$

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Problem 51

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$112^{\circ} 15^{\prime}$$

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Problem 52

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$133^{\circ} 45^{\prime}$$

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Problem 53

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$-60^{\circ} 12^{\prime}$$

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Problem 54

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$-70^{\circ} 48^{\prime}$$

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Problem 55

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$20^{\circ} 54^{\prime} 00^{\prime \prime}$$

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Problem 56

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$38^{\circ} 42^{\prime} 00^{\prime \prime}$$

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Problem 57

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$91^{\circ} 35^{\prime} 54^{\prime \prime}$$

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Problem 58

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$34^{\circ} 51^{\prime} 35^{\prime \prime}$$

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Problem 59

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$274^{\circ} 18^{\prime} 59^{\prime \prime}$$

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Problem 60

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example $4(a)$
$$165^{\circ} 51^{\prime} 09^{\prime \prime}$$

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Problem 61

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$39.25^{\circ}$$

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Problem 62

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$46.75^{\circ}$$

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Problem 63

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$126.76^{\circ}$$

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Problem 64

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$174.255^{\circ}$$

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Problem 65

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$-25.485^{\circ}$$

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Problem 66

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$-25.485^{\circ}$$

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Problem 67

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$31.4296^{\circ}$$

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Problem 68

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$59.0854^{\circ}$$

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Problem 69

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$89.9004^{\circ}$$

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Problem 70

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$102.3771^{\circ}$$

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Problem 71

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$178.5994^{\circ}$$

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Problem 72

Convert each angle measure to degrees, minutes, and seconds. Round answers to the nearest second, if applicable. See Example $4(b)$
$$122.6853^{\circ}$$

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Problem 73

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$32^{\circ}$$

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Problem 74

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$86^{\circ}$$

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Problem 75

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$26^{\circ} 30^{\prime}$$

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Problem 76

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$58^{\circ} 40^{\prime}$$

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Problem 77

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-40^{\circ}$$

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Problem 78

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-98^{\circ}$$

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Problem 79

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-125^{\circ}$$

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Problem 80

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-203^{\circ}$$

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Problem 81

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$361^{\circ}$$

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Problem 82

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$541^{\circ}$$

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Problem 83

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-361^{\circ}$$

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Problem 84

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-541^{\circ}$$

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Problem 85

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$539^{\circ}$$

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Problem 86

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$699^{\circ}$$

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Problem 87

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$850^{\circ}$$

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Problem 88

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$1000^{\circ}$$

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Problem 89

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$5280^{\circ}$$

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Problem 90

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$5280^{\circ}$$

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Problem 91

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-5280^{\circ}$$

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Problem 92

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5 .
$$-8440^{\circ}$$

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Problem 93

Give two positive and two negative angles that are coterminal with the given quadrantal angle.
$$90^{\circ}$$

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Problem 94

Give two positive and two negative angles that are coterminal with the given quadrantal angle.
$$180^{\circ}$$

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Problem 95

Give two positive and two negative angles that are coterminal with the given quadrantal angle.
$$0^{\circ}$$

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Problem 96

Give two positive and two negative angles that are coterminal with the given quadrantal angle.
$$270^{\circ}$$

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Problem 97

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$30^{\circ}$$

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Problem 98

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$45^{\circ}$$

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Problem 99

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$135^{\circ}$$

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Problem 100

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$225^{\circ}$$

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Problem 101

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$-90^{\circ}$$

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Problem 102

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$-180^{\circ}$$

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Problem 103

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$0^{\circ}$$

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Problem 104

Give an expression that generates all angles coterminal with each angle. Let n represent any integer.
$$360^{\circ}$$

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Problem 105

Explain why the answers to Exercises 103 and 104 give the same set of angles.

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Problem 106

Which two of the following are not coterminal with $r^{\circ} ?$
A. $360^{\circ}+r^{\circ}$
B. $r^{\circ}-360^{\circ}$
C. $360^{\circ}-r^{\circ}$
D. $r^{\circ}+180^{\circ}$

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Problem 107

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$75^{\circ}$$

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Problem 108

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$89^{\circ}$$

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Problem 109

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$174^{\circ}$$

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Problem 110

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$234^{\circ}$$

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Problem 111

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$300^{\circ}$$

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Problem 112

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$512^{\circ}$$

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Problem 113

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$-61^{\circ}$$

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Problem 114

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$-159^{\circ}$$

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Problem 115

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$90^{\circ}$$

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Problem 116

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$180^{\circ}$$

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Problem 117

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$-90^{\circ}$$

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Problem 118

Sketch each angle in standard position. Draw an arrow representing the correct amount of rotation. Find the measure of two other angles, one positive and one negative, that are coterminal with the given angle. Give the quadrant of each angle, if applicable.
$$-180^{\circ}$$

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Problem 119

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(-3,-3)$$

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Problem 120

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(4,-4)$$

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Problem 121

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(-3,-5)$$

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Problem 122

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(-5,2)$$

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Problem 123

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(\sqrt{2},-\sqrt{2})$$

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Problem 124

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(-2 \sqrt{2}, 2 \sqrt{2})$$

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Problem 125

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(-1, \sqrt{3})$$

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Problem 126

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(\sqrt{3}, 1)$$

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Problem 127

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(-2,2 \sqrt{3})$$

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Problem 128

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(4 \sqrt{3},-4)$$

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Problem 129

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(0,-4)$$

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Problem 130

Locate each point in a coordinate system. Draw a ray from the origin through the given point. Indicate with an arrow the angle in standard position having least positive measure. Then find the distance $r$ from the origin to the point, using the distance formula of Section 2.1
$$(0,2)$$

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Problem 131

A turntable in a shop makes 45 revolutions per min. How many revolutions does it make per second?

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Problem 132

A windmill makes 90 revolutions per min. How many revolutions does it make per second?

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Problem 133

A tire is rotating 600 times per min. Through how many degrees does a point on the edge of the tire move in $\frac{1}{2}$ sec?
(IMAGE CAN NOT COPY)

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Problem 134

An airplane propeller rotates 1000 times per min. Find the number of degrees that a point on the edge of the propeller will rotate in 1 sec.

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Problem 135

A pulley rotates through $75^{\circ}$ in 1 min. How many rotations does the pulley make in an hour?

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Problem 136

One student in a surveying class measures an angle as $74.25^{\circ},$ while another student measures the same angle as $74^{\circ} 20^{\prime} .$ Find the difference between these measurements, both to the nearest minute and to the nearest hundredth of a degree.
(IMAGE CAN NOT COPY)

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Problem 137

As a consequence of Earth's rotation, celestial objects such as the moon and the stars appear to move across the sky, rising in the east and setting in the west. As a result, if a telescope on Earth remains stationary while viewing a celestial object, the object will slowly move outside the viewing field of the telescope. For this reason, a motor is often attached to telescopes so that the telescope rotates at the same rate as Earth. Determine how long it should take the motor to turn the telescope through an angle of 1 min in a direction perpendicular to Earth's axis.

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Problem 138

Determine the measure of the angle in each point of the five-pointed star appearing on the American flag. (Hint: Inscribe the star in a circle, and use the following theorem from geometry: $A n$ angle whose vertex lies on the circumference of a circle is equal to half the central angle that cuts off the same arc. See the figure.)
(FIGURE CAN NOT COPY)

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