Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 6

Trigonometric Functions: Right Triangle Approach - all with Video Answers

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Section 1

Angle Measure

Problem 1

(a) The radian measure of an angle $\theta$ is the length of the _________ that subtends the angle in a circle of radius______________
(b) To convert degrees to radians, we multiply by ___________
(c) To convert radians to degrees, we multiply by _____________

Emily B.

Emily B.

Numerade Educator

Problem 2

A central angle $\theta$ is drawn in a circle of radius $r$
(a) The length of the arc subtended by $\theta$ is $s=$ _______________
(b) The area of the sector with central angle $\theta$ is $A=$ _____________

Emily B.

Numerade Educator

Problem 3

Find the radian measure of the angle with the given degree measure.
$$
72^{\circ}
$$

Joseph L.

Numerade Educator

Problem 4

Find the radian measure of the angle with the given degree measure.
$$
54^{\circ}
$$

Joseph L.

Numerade Educator

Problem 5

Find the radian measure of the angle with the given degree measure.
$$
-45^{\circ}
$$

Joseph L.

Numerade Educator

Problem 6

Find the radian measure of the angle with the given degree measure.
$$
-60^{\circ}
$$

Joseph L.

Numerade Educator

Problem 7

Find the radian measure of the angle with the given degree measure.
$$
-75^{\circ}
$$

Joseph L.

Numerade Educator

Problem 8

Find the radian measure of the angle with the given degree measure.
$$
-300^{\circ}
$$

Joseph L.

Numerade Educator

Problem 9

Find the radian measure of the angle with the given degree measure.
$$
1080^{\circ}
$$

Joseph L.

Numerade Educator

Problem 10

Find the radian measure of the angle with the given degree measure.
$$
3960^{\circ}
$$

Joseph L.

Numerade Educator

Problem 11

Find the radian measure of the angle with the given degree measure.
$$
96^{\circ}
$$

Joseph L.

Numerade Educator

Problem 12

Find the radian measure of the angle with the given degree measure.
$$
15^{\circ}
$$

Joseph L.

Numerade Educator

Problem 13

Find the radian measure of the angle with the given degree measure.
$$
7.5^{\circ}
$$

Joseph L.

Numerade Educator

Problem 14

Find the radian measure of the angle with the given degree measure.
$$
202.5^{\circ}
$$

Joseph L.

Numerade Educator

Problem 15

Find the degree measure of the angle with the given radian measure.
$$
\frac{7 \pi}{6}
$$

Joseph L.

Numerade Educator

Problem 16

Find the degree measure of the angle with the given radian measure.
$$
\frac{11 \pi}{3}
$$

Joseph L.

Numerade Educator

Problem 17

Find the degree measure of the angle with the given radian measure.
$$
-\frac{5 \pi}{4}
$$

Joseph L.

Numerade Educator

Problem 18

Find the degree measure of the angle with the given radian measure.
$$
-\frac{3 \pi}{2}
$$

Anthony P.

Numerade Educator

Problem 19

Find the degree measure of the angle with the given radian measure.
$$
3
$$

Joseph L.

Numerade Educator

Problem 20

Find the degree measure of the angle with the given radian measure.
$$
-2
$$

Joseph L.

Numerade Educator

Problem 21

Find the degree measure of the angle with the given radian measure.
$$
-1.2
$$

Joseph L.

Numerade Educator

Problem 22

Find the degree measure of the angle with the given radian measure.
$$
3.4
$$

Joseph L.

Numerade Educator

Problem 23

Find the degree measure of the angle with the given radian measure.
$$
\frac{\pi}{10}
$$

Martha R.

Numerade Educator

Problem 24

Find the degree measure of the angle with the given radian measure.
$$
\frac{5 \pi}{18}
$$

Anthony P.

Numerade Educator

Problem 25

Find the degree measure of the angle with the given radian measure.
$$
-\frac{2 \pi}{15}
$$

Joseph L.

Numerade Educator

Problem 26

Find the degree measure of the angle with the given radian measure.
$-\frac{13 \pi}{12}$

Joseph L.

Numerade Educator

Problem 27

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
50^{\circ}
$$

Joseph L.

Numerade Educator

Problem 28

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
135^{\circ}
$$

Joseph L.

Numerade Educator

Problem 29

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
\frac{3 \pi}{4}
$$

Joseph L.

Numerade Educator

Problem 30

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
\frac{11 \pi}{6}
$$

Joseph L.

Numerade Educator

Problem 31

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
-\frac{\pi}{4}
$$

Joseph L.

Numerade Educator

Problem 32

The measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
-45^{\circ}
$$

Joseph L.

Numerade Educator

Problem 33

The measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
70^{\circ}, \quad 430^{\circ}
$$

Joseph L.

Numerade Educator

Problem 34

The measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
-30^{\circ}, \quad 330^{\circ}
$$

Joseph L.

Numerade Educator

Problem 35

The measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
\frac{5 \pi}{6}, \frac{17 \pi}{6}
$$

Emily B.

Numerade Educator

Problem 36

The measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
\frac{32 \pi}{3}, \frac{11 \pi}{3}
$$

Joseph L.

Numerade Educator

Problem 37

The measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
155^{\circ}, \quad 875^{\circ}
$$

Joseph L.

Numerade Educator

Problem 38

The measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
50^{\circ}, \quad 340^{\circ}
$$

Joseph L.

Numerade Educator

Problem 39

Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
733^{\circ}
$$

Joseph L.

Numerade Educator

Problem 40

Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
361^{\circ}
$$

Joseph L.

Numerade Educator

Problem 41

Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
1110^{\circ}
$$

Joseph L.

Numerade Educator

Problem 42

Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-100^{\circ}
$$

Emily B.

Numerade Educator

Problem 43

Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-800^{\circ}
$$

Emily B.

Numerade Educator

Problem 44

Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
1270^{\circ}
$$

Emily B.

Numerade Educator

Problem 45

Find an angle between 0 and 2p that is coterminal with the given angle.
$$
\frac{17 \pi}{6}
$$

Joseph L.

Numerade Educator

Problem 46

Find an angle between 0 and 2p that is coterminal with the given angle.
$$
-\frac{7 \pi}{3}
$$

Joseph L.

Numerade Educator

Problem 47

Find an angle between 0 and 2p that is coterminal with the given angle.
$$87 \pi$$

Joseph L.

Numerade Educator

Problem 48

Find an angle between 0 and 2p that is coterminal with the given angle.
$$
10
$$

Joseph L.

Numerade Educator

Problem 49

Find an angle between 0 and 2p that is coterminal with the given angle.
$$
\frac{17 \pi}{4}
$$

Joseph L.

Numerade Educator

Problem 50

Find an angle between 0 and 2p that is coterminal with the given angle.
$$
\frac{51 \pi}{2}
$$

Joseph L.

Numerade Educator

Problem 51

Find the length of the arc $s$ in the figure.

Julian W.

Numerade Educator

Problem 52

Find the angle $\theta$ in the figure.

Joseph L.

Numerade Educator

Problem 53

Find the radius $r$ of the circle in the figure.

Julian W.

Numerade Educator

Problem 54

Find the length of an arc that subtends a central angle of $45^{\circ}$ in a circle of radius $10 \mathrm{m} .$

Joseph L.

Numerade Educator

Problem 55

Find the length of an arc that subtends a central angle of 2 rad in a circle of radius $2 \mathrm{mi} .$

Pawan Y.

Numerade Educator

Problem 56

A central angle $\theta$ in a circle of radius 5 $\mathrm{m}$ is subtended by an arc of length $6 \mathrm{m} .$ Find the measure of $\theta$ in degrees and in radians.

Joseph L.

Numerade Educator

Problem 57

An arc of length 100 $\mathrm{m}$ subtends a central angle $\theta$ in a circle of radius 50 $\mathrm{m}$ . Find the measure of $\theta$ in degrees and in radians.

Joseph L.

Numerade Educator

Problem 58

A circular arc of length 3 $\mathrm{ft}$ subtends a central angle of $25^{\circ}$ . Find the radius of the circle.

Joseph L.

Numerade Educator

Problem 59

Find the radius of the circle if an arc of length 6 $\mathrm{m}$ on the circle subtends a central angle of $\pi / 6 \mathrm{rad}$ .

Joseph L.

Numerade Educator

Problem 60

Find the radius of the circle if an arc of length 4 $\mathrm{ft}$ on the circle subtends a central angle of $135^{\circ} .$

Joseph L.

Numerade Educator

Problem 61

Find the area of the sector shown in each figure.

CC

Charles C.

Numerade Educator

Problem 62

Find the radius of each circle if the area of the sector is 12 .

DL

Debra L.

Numerade Educator

Problem 63

Find the area of a sector with central angle 1 rad in a circle of radius $10 \mathrm{m} .$

Joseph L.

Numerade Educator

Problem 64

A sector of a circle has a central angle of $60^{\circ} .$ Find the area of the sector if the radius of the circle is $3 \mathrm{mi} .$

Joseph L.

Numerade Educator

Problem 65

The area of a sector of a circle with a central angle of 2 rad is $16 \mathrm{m}^{2} .$ Find the radius of the circle.

Joseph L.

Numerade Educator

Problem 66

A sector of a circle of radius 24 $\mathrm{mi}$ has an area of 288 $\mathrm{mi}^{2}$ . Find the central angle of the sector.

Joseph L.

Numerade Educator

Problem 67

The area of a circle is $72 \mathrm{cm}^{2} .$ Find the area of a sector of this circle that subtends a central angle of $\pi / 6 \mathrm{rad}$ .

Joseph L.

Numerade Educator

Problem 68

Three circles with radii $1,2,$ and 3 ft are externally tangent to one another, as shown in the figure. Find the area of the sector of the circle of radius 1 that is cut off by the line segments joining the center of that circle to the centers of the other two circles.

Joseph L.

Numerade Educator

Problem 69

Travel Distance A car's wheels are 28 in. in diameter. How far (in miles) will the car travel if its wheels revolve $10,000$ times without slipping?

Joseph L.

Numerade Educator

Problem 70

Wheel Revolutions How many revolutions will a car wheel of diameter 30 in. make as the car travels a distance of one mile?

Emily B.

Numerade Educator

Problem 71

Latitudes Pittsburgh, Pennsylvania, and Miami, Florida, lie approximately on the same meridian. Pittsburgh has a latitude of $40.5^{\circ} \mathrm{N},$ and Miami has a latitude of $25.5^{\circ} \mathrm{N} .$ Find the distance between these two cities. (The radius of the earth is 3960 $\mathrm{mi}$ .)

Arulmozhi T.

Numerade Educator

Problem 72

Latitudes Memphis, Tennessee, and New Orleans, Louisiana, lie approximately on the same meridian. Memphis has a latitude of $35^{\circ} \mathrm{N}$ , and New Orleans has a latitude of $30^{\circ} \mathrm{N}$ . Find the distance between these two cities. (The radius of the earth is 3960 $\mathrm{mi}$ .)

Emily B.

Numerade Educator

Problem 73

Orbit of the Earth Find the distance that the earth travels in one day in its path around the sun. Assume that a year has 365 days and that the path of the earth around the sun is a circle of radius 93 million miles. IThe path of the earth around the sun is actually an ellipse with the sun at one focus (see Section $11.2 ) .$ This ellipse, however, has very small eccentricity, so it is nearly circular.]

YS

Yuankun S.

Numerade Educator

Problem 74

Circumference of the Earth The Greek mathematician Eratosthenes (ca. $276-195 \mathrm{B}$ . C. ) measured the circumference of the earth from the following observations. He noticed that on a certain day the sun shone directly down a deep well in Syene (modern Aswan). At the same time in Alexandria, 500 miles north (on the same meridian), the rays of the sun shone at an angle of $7.2^{\circ}$ to the zenith. Use this information and the figure to find the radius and circumference of the earth.

Anthony P.

Numerade Educator

Problem 75

Nautical Miles Find the distance along an arc on the sur- face of the earth that subtends a central angle of 1 minute $\left(1 \text { minute }=\frac{1}{60} \text { degree). This distance is called a nautical mile. }\right.$ (The radius of the earth is 3960 $\mathrm{mi}$ .)

AR

Aakash R.

Numerade Educator

Problem 76

Irrigation An irrigation system uses a straight sprinkler pipe 300 $\mathrm{ft}$ long that pivots around a central point as shown. Due to an obstacle the pipe is allowed to pivot through $280^{\circ}$ only. Find the area irrigated by this system.

Joseph L.

Numerade Educator

Problem 77

Windshield Wipers The top and bottom ends of a wind-shield wiper blade are 34 in. and 14 in, respectively, from the pivot point. While in operation, the wiper sweeps through $135^{\circ} .$ Find the area swept by the blade.

Joseph L.

Numerade Educator

Problem 78

The Tethered Cow A cow is tethered by a $100-$ ft rope to the inside corner of an L-shaped building, as shown in the figure. Find the area that the cow can graze.

Emily B.

Numerade Educator

Problem 79

Fan A ceiling fan with 16 -in. blades rotates at 45 $\mathrm{rpm}$ .
(a) Find the angular speed of the fan in rad/min.
(b) Find the linear speed of the tips of the blades in in./min.

Emily B.

Numerade Educator

Problem 80

Radial Saw A radial saw has a blade with a 6 -in. radius. Suppose that the blade spins at 1000 $\mathrm{rpm}$ .
(a) Find the angular speed of the blade in rad/min.
(b) Find the linear speed of the sawteeth in $\mathrm{ft} / \mathrm{s}$ .

Emily B.

Numerade Educator

Problem 81

Winch A winch of radius 2 $\mathrm{ft}$ is used to lift heavy loads. If the winch makes 8 revolutions every 15 $\mathrm{s}$ , find the speed at which the load is rising.

Ashley H.

Numerade Educator

Problem 82

Speed of a Car The wheels of a car have radius 11 in. and are rotating at $600 \mathrm{rpm} .$ Find the speed of the car in mi/h.

Emily B.

Numerade Educator

Problem 83

Speed at the Equator The earth rotates about its axis once every 23 $\mathrm{h} 56 \mathrm{min} 4 \mathrm{s}$ , and the radius of the earth is 3960 $\mathrm{mi}$ . Find the linear speed of a point on the equator in mi/h.

Emily B.

Numerade Educator

Problem 84

Truck Wheels A truck with 48 -in.-diameter wheels is traveling at 50 $\mathrm{mi} / \mathrm{h}$ .
(a) Find the angular speed of the wheels in rad/min.
(b) How many revolutions per minute do the wheels make?

Emily B.

Numerade Educator

Problem 85

Speed of a Current To measure the speed of a current, scientists place a paddle wheel in the stream and observe the rate at which it rotates. If the paddle wheel has radius 0.20 $\mathrm{m}$ and rotates at $100 \mathrm{rpm},$ find the speed of the current in $\mathrm{m} / \mathrm{s}$ .

Emily B.

Numerade Educator

Problem 86

Bicycle Wheel The sprockets and chain of a bicycle are shown in the figure. The pedal sprocket has a radius of 4 in. the wheel sprocket a radius of 2 in. and the wheel a radius of 13 in. The cyclist pedals at 40 $\mathrm{rpm}$ .
(a) Find the angular speed of the wheel sprocket.
(b) Find the speed of the bicycle. (Assume that the wheel turns at the same rate as the wheel sprocket.)

Emily B.

Numerade Educator

Problem 87

Conical Cup A conical cup is made from a circular piece of paper with radius 6 $\mathrm{cm}$ by cutting out a sector and joining the edges as shown on the next page. Suppose $\theta=5 \pi / 3 .$
(a) Find the circumference $C$ of the opening of the cup.
(b) Find the radius $r$ of the opening of the cup. $[\text { Hint: Use }$ $C=2 \pi r . ]$
(c) Find the height $h$ of the cup. $[\text { Hint: Use the Pythagorean }$ Theorem.]
(d) Find the volume of the cup.

Check back soon!

Problem 88

Conical Cup In this exercise we find the volume of the conical cup in Exercise 87 for any angle $\theta$ .
(a) Follow the steps in Exercise 87 to show that the volume of the cup as a function of $\theta$ is
$$
V(\theta)=\frac{9}{\pi^{2}} \theta^{2} \sqrt{4 \pi^{2}-\theta^{2}}, \quad 0<\theta<2 \pi
$$
(b) Graph the function $V$ .
(c) For what angle $\theta$ is the volume of the cup a maximum?

Anthony P.

Numerade Educator

Problem 89

Different Ways of Measuring Angles The custom of measuring angles using degrees, with $360^{\circ}$ in a circle, dates back to the ancient Babylonians, who used a number system based on groups of $60 .$ Another system of measuring angles divides the circle into 400 units, called grads. In this system a right angle is circle into 400 units, called grads. In this system a right angle is 100 grad, so this fits in with our base 10 number system. Write a short essay comparing the advantages and disadvantages of these two systems and the radian system of measuring angles. Which system do you prefer? Why?

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Problem 90

Clocks and Angles In one hour, the minute hand on a clock moves through a complete circle, and the hour hand moves through $\frac{1}{12}$ of a circle. Through how many radians do the minute and the hour hand move between $1 : 00$ P.M. and $6 : 45$ P.M. (on the same day)?

Arulmozhi T.

Numerade Educator