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Algebra and Trigonometry

James Stewart, Lothar Redlin, Saleem Watson

Chapter 6

Trigonometric Functions of Angles - all with Video Answers

Educators

+ 4 more educators

Section 1

Angle Measure

01:29

Problem 1

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
72^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:34

Problem 2

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
54^{\circ}
$$

Willis James
Willis James
Numerade Educator
01:58

Problem 3

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
-45^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:35

Problem 4

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
-60^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:54

Problem 5

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
-75^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:45

Problem 6

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
-300^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:30

Problem 7

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
1080^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:30

Problem 8

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
3960^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:34

Problem 9

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
96^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:29

Problem 10

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
15^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:53

Problem 11

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
7.5^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
02:28

Problem 12

$1-12$ . Find the radian measure of the angle with the given degree measure.
$$
202.5^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:13

Problem 13

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
\frac{7 \pi}{6}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:03

Problem 14

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
\frac{11 \pi}{3}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:13

Problem 15

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
-\frac{5 \pi}{4}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:16

Problem 16

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
-\frac{3 \pi}{2}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
02:11

Problem 17

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
3
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:49

Problem 18

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
-2
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:57

Problem 19

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
-1.2
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:36

Problem 20

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
3.4
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:19

Problem 21

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
\frac{\pi}{10}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:19

Problem 22

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
\frac{5 \pi}{18}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:15

Problem 23

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
-\frac{2 \pi}{15}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:12

Problem 24

$13-24$ . Find the degree measure of the angle with the given radian measure.
$$
-\frac{13 \pi}{12}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
02:00

Problem 25

$25-30$ me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
50^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
02:16

Problem 26

$25-30$ me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
135^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
02:28

Problem 27

$25-30$ me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
\frac{3 \pi}{4}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
03:07

Problem 28

$25-30$ me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
\frac{11 \pi}{6}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
02:20

Problem 29

$25-30$ me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
-\frac{\pi}{4}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:56

Problem 30

$25-30$ me measure of an angle in standard position is given. Find two positive angles and two negative angles that are coterminal with the given angle.
$$
-45^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:22

Problem 31

$31-36$ me measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
70^{\circ}, 430^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:54

Problem 32

$31-36$ me measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
-30^{\circ}, \quad 330^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:47

Problem 33

$31-36$ me measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
\frac{5 \pi}{6}, \frac{17 \pi}{6}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:59

Problem 34

$31-36$ me measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
\frac{32 \pi}{3}, \frac{11 \pi}{3}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:50

Problem 35

$31-36$ me measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
155^{\circ}, \quad 875^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:22

Problem 36

$31-36$ me measures of two angles in standard position are given. Determine whether the angles are coterminal.
$$
50^{\circ}, \quad 340^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:33

Problem 37

$37-42$ me Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
733^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:09

Problem 38

$37-42$ me Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
361^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:45

Problem 39

$37-42$ me Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
1110^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:20

Problem 40

$37-42$ me Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-100^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:43

Problem 41

$37-42$ me Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
-800^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:32

Problem 42

$37-42$ me Find an angle between $0^{\circ}$ and $360^{\circ}$ that is coterminal with the given angle.
$$
1270^{\circ}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:34

Problem 43

$43-48=$ Find an angle between 0 and 2$\pi$ that is coterminal with the given angle.
$$
\frac{17 \pi}{6}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:59

Problem 44

$43-48=$ Find an angle between 0 and 2$\pi$ that is coterminal with the given angle.
$$
-\frac{7 \pi}{3}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:22

Problem 45

$43-48=$ Find an angle between 0 and 2$\pi$ that is coterminal with the given angle.
$$
87 \pi
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
02:25

Problem 46

$43-48=$ Find an angle between 0 and 2$\pi$ that is coterminal with the given angle.
$$
10
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:15

Problem 47

$43-48=$ Find an angle between 0 and 2$\pi$ that is coterminal with the given angle.
$$
\frac{17 \pi}{4}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
01:32

Problem 48

$43-48=$ Find an angle between 0 and 2$\pi$ that is coterminal with the given angle.
$$
\frac{51 \pi}{2}
$$

Arulmozhi T
Arulmozhi T
Numerade Educator
00:35

Problem 49

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

Julian Wong
Julian Wong
Numerade Educator
00:51

Problem 50

Find the angle $\theta$ in the figure.

Joseph Lentino
Joseph Lentino
Numerade Educator
00:42

Problem 51

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

Julian Wong
Julian Wong
Numerade Educator
01:22

Problem 52

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

Joseph Lentino
Joseph Lentino
Numerade Educator
01:28

Problem 53

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

Arulmozhi T
Arulmozhi T
Numerade Educator
01:12

Problem 54

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 Lentino
Joseph Lentino
Numerade Educator
02:56

Problem 55

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.

Stephen Hobbs
Stephen Hobbs
Numerade Educator
03:25

Problem 56

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

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:16

Problem 57

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 Lentino
Joseph Lentino
Numerade Educator
01:51

Problem 58

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 Lentino
Joseph Lentino
Numerade Educator
02:40

Problem 59

Find the area of the sector shown in each figure.

Charles Carter
Charles Carter
Numerade Educator
15:42

Problem 60

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

DL
Debra Leonard
Numerade Educator
01:53

Problem 61

Find the area of a sector with central angle 1 rad in a circle of radius 10 m.

Linda Hand
Linda Hand
Numerade Educator
01:10

Problem 62

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 Lentino
Joseph Lentino
Numerade Educator
View

Problem 63

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.

Daniel Carr
Daniel Carr
Numerade Educator
01:57

Problem 64

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.

Suman Saurav Thakur
Suman Saurav Thakur
Numerade Educator
01:42

Problem 65

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 $\pi / 6$ rad.

Joseph Lentino
Joseph Lentino
Numerade Educator
07:55

Problem 66

Three circles with radii $1,2,$ and 3 $\mathrm{ft}$ are externally tangent to one another, as shown in the figure on the next page. 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.

Arulmozhi T
Arulmozhi T
Numerade Educator
02:44

Problem 67

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 Lentino
Joseph Lentino
Numerade Educator
01:49

Problem 68

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

Emily Burns
Emily Burns
Numerade Educator
05:38

Problem 69

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

Arulmozhi T
Arulmozhi T
Numerade Educator
01:52

Problem 70

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

Emily Burns
Emily Burns
Numerade Educator
00:24

Problem 71

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. [The path of the earth around the sun is actually miles. [The path of the earth around the sun is actually an ellipse, hith the sun at one focus (see Section 11.2$)$ . This ellipse, however, has very small eccentricity, so it is nearly circular.

YS
Yuankun Song
Numerade Educator
02:46

Problem 72

Circumference of the Earth The Greek mathematician Eratosthenes (ca. $276-195$ 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 Pisaturo
Anthony Pisaturo
Numerade Educator
01:46

Problem 73

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

AR
Aakash Ramachandran
Numerade Educator
02:02

Problem 74

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 Lentino
Joseph Lentino
Numerade Educator
02:33

Problem 75

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

Joseph Lentino
Joseph Lentino
Numerade Educator
03:34

Problem 76

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 Burns
Emily Burns
Numerade Educator
03:00

Problem 77

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.

Emily Burns
Emily Burns
Numerade Educator
04:09

Problem 78

Fan A ceiling fan with 16-in. blades rotates at 45 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.

Bobby Barnes
Bobby Barnes
University of North Texas
02:17

Problem 79

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

Ashly Sunny
Ashly Sunny
Numerade Educator
02:51

Problem 80

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

Emily Burns
Emily Burns
Numerade Educator
04:15

Problem 81

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

Emily Burns
Emily Burns
Numerade Educator
02:47

Problem 82

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

Emily Burns
Emily Burns
Numerade Educator
01:33

Problem 83

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 m and rotates at 100 rpm, find the speed of the current in m/s.

Emily Burns
Emily Burns
Numerade Educator
07:45

Problem 84

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 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.)

Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsohdsno3/5Wc5Bsgjhyqjgxswzij15Z06 Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsoec1F7Nikxndin/Owbntbjji9Jzcznki
Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsohdsno3/5Wc5Bsgjhyqjgxswzij15Z06 Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsoec1F7Nikxndin/Owbntbjji9Jzcznki
Numerade Educator
09:14

Problem 85

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. 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. [Hint: Use $C=2 \pi r . ]$
(c) Find the height $h$ of the cup. IHint: Use the Pythagorean Theorem.]
(d) Find the volume of the cup.

Arulmozhi T
Arulmozhi T
Numerade Educator
01:35

Problem 86

Conical Cup In this exercise we find the volume of the
conical cup in Exercise 85 for any angle $\theta .$
(a) Follow the steps in Exercise 85 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 Pisaturo
Anthony Pisaturo
Numerade Educator
00:44

Problem 87

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 100 grad, so this fits in with our base 10 number system. Write a short essay comparing the advantages and disad- vantages of these two systems and the radian system of measuring angles. Which system do you prefer?

Katelyn Chen
Katelyn Chen
Numerade Educator
06:16

Problem 88

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 $1 : 00 \mathrm{P.M.}$ and $6 : 45$ P.M. (on the same day)?

Arulmozhi T
Arulmozhi T
Numerade Educator