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Precalculus: Graphical, Numerical, Algebraic

Franklin D. Demana,Bert K. Waits,Gregory D. Foley,Daniel Kennedy

Chapter 4

Trigonometric Functions - all with Video Answers

Educators

Section 1

Angles and Their Measures

01:25

Problem 1

In Exercises $1-4,$ convert from DMS to decimal form.
$$
23^{\circ} 12^{\prime}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
03:56

Problem 2

In Exercises $1-4,$ convert from DMS to decimal form.
$$
35^{\circ} 24^{\prime}
$$

Eric Tung
Eric Tung
Numerade Educator
01:54

Problem 3

In Exercises $1-4,$ convert from DMS to decimal form.
$$
118^{\circ} 44^{\prime} 15^{\prime \prime}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
04:28

Problem 4

In Exercises $1-4,$ convert from DMS to decimal form.
$$
48^{\circ} 30^{\prime} 36^{\prime \prime}
$$

Eric Tung
Eric Tung
Numerade Educator
00:49

Problem 5

In Exercises $5-8,$ convert from decimal form to degrees, minutes, seconds (DMS).
$$
21.2^{\circ}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
03:55

Problem 6

In Exercises $5-8,$ convert from decimal form to degrees, minutes, seconds (DMS).
$$
49.7^{\circ}
$$

Eric Tung
Eric Tung
Numerade Educator
01:00

Problem 7

In Exercises $5-8,$ convert from decimal form to degrees, minutes, seconds (DMS).
$$
118.32^{\circ}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
05:41

Problem 8

In Exercises $5-8,$ convert from decimal form to degrees, minutes, seconds (DMS).
$$
99.37^{\circ}
$$

Eric Tung
Eric Tung
Numerade Educator
00:54

Problem 9

In Exercises $9-16,$ convert from DMS to radians.
$$
60^{\circ}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:49

Problem 10

In Exercises $9-16,$ convert from DMS to radians.
$$
90^{\circ}
$$

Eric Tung
Eric Tung
Numerade Educator
00:48

Problem 11

In Exercises $9-16,$ convert from DMS to radians.
$$
120^{\circ}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:05

Problem 12

In Exercises $9-16,$ convert from DMS to radians.
$$
150^{\circ}
$$

Eric Tung
Eric Tung
Numerade Educator
02:37

Problem 13

In Exercises $9-16,$ convert from DMS to radians.
$$
71.72^{\circ}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:26

Problem 14

In Exercises $9-16,$ convert from DMS to radians.
$$
11.83^{\circ}
$$

Eric Tung
Eric Tung
Numerade Educator
01:29

Problem 15

In Exercises $9-16,$ convert from DMS to radians.
$$
61^{\circ} 24^{\prime}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
03:38

Problem 16

In Exercises $9-16,$ convert from DMS to radians.
$$
75^{\circ} 30^{\prime}
$$

Eric Tung
Eric Tung
Numerade Educator
00:38

Problem 17

In Exercises $17-24,$ convert from radians to degrees.
$$
\pi / 6
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:06

Problem 18

In Exercises $17-24,$ convert from radians to degrees.
$$
\pi / 4
$$

Eric Tung
Eric Tung
Numerade Educator
00:59

Problem 19

In Exercises $17-24,$ convert from radians to degrees.
$$
\pi / 10
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:36

Problem 20

In Exercises $17-24,$ convert from radians to degrees.
$$
3 \pi / 5
$$

Eric Tung
Eric Tung
Numerade Educator
00:54

Problem 21

In Exercises $17-24,$ convert from radians to degrees.
$$
7 \pi / 9
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:19

Problem 22

In Exercises $17-24,$ convert from radians to degrees.
$$
13 \pi / 20
$$

Eric Tung
Eric Tung
Numerade Educator
01:04

Problem 23

In Exercises $17-24,$ convert from radians to degrees.
$$
2
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:15

Problem 24

In Exercises $17-24,$ convert from radians to degrees.
$$
1.3
$$

Eric Tung
Eric Tung
Numerade Educator
01:03

Problem 25

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Joshua Eastwood
Joshua Eastwood
Numerade Educator
00:54

Problem 26

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Eric Tung
Eric Tung
Numerade Educator
01:14

Problem 27

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Joshua Eastwood
Joshua Eastwood
Numerade Educator
01:11

Problem 28

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Eric Tung
Eric Tung
Numerade Educator
00:46

Problem 29

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Joshua Eastwood
Joshua Eastwood
Numerade Educator
01:22

Problem 30

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Eric Tung
Eric Tung
Numerade Educator
01:48

Problem 31

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:35

Problem 32

In Exercises $25-32$ , use the appropriate arc length formula to find the missing information.
Table cannot copy

Eric Tung
Eric Tung
Numerade Educator
01:41

Problem 33

In Exercises 33 and $34,$ a central angle $\theta$ intercepts arcs $s_{1}$ and $s_{2}$ on two concentric circles with radii $r_{1}$ and $r_{2},$ respectively. Find the missing information.
Table cannot copy

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:40

Problem 34

In Exercises 33 and $34,$ a central angle $\theta$ intercepts arcs $s_{1}$ and $s_{2}$ on two concentric circles with radii $r_{1}$ and $r_{2},$ respectively. Find the missing information.
Table cannot copy

Eric Tung
Eric Tung
Numerade Educator
01:35

Problem 35

To the nearest inch, find the perimeter of a 10 -degree sector cut from a circular disc of radius 11 inches.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
01:44

Problem 36

A 100 -degree arc of a circle has a length of 7 $\mathrm{cm} .$ To the nearest centimeter, what is the radius of the circle?

Eric Tung
Eric Tung
Numerade Educator
04:18

Problem 37

It takes ten identical pieces to form a circular track for a pair of toy racing cars. If the inside arc of each piece is 3.4 inches shorter than the outside arc, what is the width of the track?

Joshua Eastwood
Joshua Eastwood
Numerade Educator
03:20

Problem 38

The concentric circles on an archery target are 6 inches apart. The inner circle (red) has a perimeter of 37.7 inches. What is the perimeter of the next-largest (yellow) circle?

Eric Tung
Eric Tung
Numerade Educator
02:23

Problem 39

Exercises $39-42$ refer to the 16 compass bearings shown. North corresponds to an angle of $0^{\circ},$ and other angles are measured clockwise from north.
Compass Reading Find the angle in degrees that describes the compass bearing.
(a) $\mathrm{NE}$ (northeast)
(b) NNE (north-northeast)
(c) WSW (west-southwest)

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:30

Problem 40

Exercises $39-42$ refer to the 16 compass bearings shown. North corresponds to an angle of $0^{\circ},$ and other angles are measured clockwise from north.
Compass Reading Find the angle in degrees that describes the compass bearing.
(a) SSW (south-southwest)
(b) WNW (west-northwest)
(c) NNW ( north-northwest)

Eric Tung
Eric Tung
Numerade Educator
01:33

Problem 41

Exercises $39-42$ refer to the 16 compass bearings shown. North corresponds to an angle of $0^{\circ},$ and other angles are measured clockwise from north.
Compass Reading Which compass direction is closest to a bearing of $121^{\circ}$ ?

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:26

Problem 42

Exercises $39-42$ refer to the 16 compass bearings shown. North corresponds to an angle of $0^{\circ},$ and other angles are measured clockwise from north.
Compass Reading Which compass direction is closest to a bearing of $219^{\circ}$ ?

Eric Tung
Eric Tung
Numerade Educator
01:37

Problem 43

Navigation Two Coast Guard patrol boats leave Cape May at the same time. One travels with a bearing of $42^{\circ} 30^{\prime}$ and the other with a bearing of $52^{\circ} 12^{\prime} .$ If they travel at the same speed, approximately how far apart will they be when they are 25 statute miles from Cape May?

Kratika Bhadauria
Kratika Bhadauria
Numerade Educator
13:54

Problem 44

Automobile Design Table 4.1 shows the size specifications for the tires that come as standard equipment on three different 2009 American vehicles.
$$
\begin{array}{lll}{\text { Vehicle }} & {\text { Tire Type }} & {\text { Tire Diameter }} \\ \hline \text { Ford Taurus } & {215 / 60-17} & {27.2 \text { inches }} \\ {\text { Dodge Charger RT }} & {225 / 60-18} & {28.6 \text { inches }} \\ {\text { Mercury Mariner }} & {235 / 70-16} & {29.0 \text { inches }}\end{array}
$$
(a) Find the speed of each vehicle in mph when the wheels are turning at 800 revolutions per minute.
(b) Compared to the Mercury Mariner, how many more revolutions must the tire of the Ford Taurus make in order to travel a mile?
(c) Writing to Learn It is unwise (and in some cases illegal) to equip a vehicle with wheels of a larger diameter than those for which it was designed. If a 2009 Ford Taurus were equipped with 29-inch tires, how would it affect the odometer (which measures mileage) and speedometer readings?

Eric Tung
Eric Tung
Numerade Educator
01:49

Problem 45

Bicycle Racing Cathy Nguyen races on a bicycle with 13 -inch-radius wheels. When she is traveling at a speed of 44 ft sec, how many revolutions per minute are her wheels making?

Joshua Eastwood
Joshua Eastwood
Numerade Educator
03:57

Problem 46

Tire Sizing The numbers in the "tire type" column in Exercise 44 give the size of the tire in the P-metric system. Each number is of the form $W / R-D,$ where $W$ is the width of the tire in millimeters, $R / 100$ is the ratio of the sidewall $(S)$ of the tire to its width $W,$ and $D$ is the diameter (in inches) of the wheel without the tire.
(a) Show that $S=W R / 100$ millimeters $=W R / 2540$ inches.
(b) The tire diameter is $D+2 S .$ Derive a formula for the tire diameter that involves only the variables $D, W,$ and $R .$
(c) Use the formula in part (b) to verify the tire diameters in Exercise $44 .$ Then find the tire diameter for the 2009 Honda Ridgeline, which comes with $245 / 65-17$ tires.

Eric Tung
Eric Tung
Numerade Educator
02:53

Problem 47

Tool Design A radial arm saw has a circular cutting blade with a diameter of 10 inches. It spins at 2000 $\mathrm{rpm} .$ If there are 12 cutting teeth per inch on the cutting blade, how many teeth cross the cutting surface each second?

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:13

Problem 48

Navigation Sketch a diagram of a ship on the given course.
(a) $35^{\circ} \quad$ (b) $128^{\circ} \quad$ (e) $310^{\circ}$

Eric Tung
Eric Tung
Numerade Educator
01:50

Problem 49

Navigation The captain of the tourist boat Julia out of Oak Harbor follows a $38^{\circ}$ course for 2 miles and then changes to a $47^{\circ}$ course for the next 4 miles. Draw a sketch of this trip.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:21

Problem 50

Navigation Points $A$ and $B$ are 257 nautical miles apart. How far apart are $A$ and $B$ in statute miles?

Eric Tung
Eric Tung
Numerade Educator
01:14

Problem 51

Navigation Points $C$ and $D$ are 895 statute miles apart. How far apart are $C$ and $D$ in nautical miles?

Joshua Eastwood
Joshua Eastwood
Numerade Educator
04:29

Problem 52

Designing a Sports Complex Example 4 describes how lanes 1 and 2 compare in length around one turn of a track. Find the differences in the lengths of these lanes around one turn of the same track.
(a) Lanes 5 and 6$\quad$ (b) Lanes 1 and 6

Eric Tung
Eric Tung
Numerade Educator
01:20

Problem 53

Mechanical Engineering A simple pulley with the given radius $r$ used to lift heavy objects is positioned 10 feet above ground level. Given that the pulley rotates $\theta^{\circ},$ determine the height to which the object is lifted.
(a) $r=4$ in. $\theta=720^{\circ} \qquad$ (b) $r=2 \mathrm{ft}, \theta=180^{\circ}$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:29

Problem 54

Foucault Pendulum In 1851 the French physicist Jean Foucault used a pendulum to demonstrate the Earth's rotation. There are now over 30 Foucault pendulum displays in the United States. The Foucault pendulum at the Smithsonian Institution in Washington, DC, consists of a large brass ball suspended by a thin 52 -foot cable. If the ball swings through an angle of $1^{\circ},$ how far does it travel?

Eric Tung
Eric Tung
Numerade Educator
02:23

Problem 55

Group Aetivity Air Conditioning Belt The belt on an automobile air conditioner connects metal wheels with radii $r=4 \mathrm{cm}$ and $R=7 \mathrm{cm} .$ The angular speed of the larger wheel is 120 $\mathrm{rpm} .$
(a) What is the angular speed of the larger wheel in radians per second?
(b) What is the linear speed of the belt in centimeters per second?
(c) What is the angular speed of the smaller wheel in radians per second?

Joshua Eastwood
Joshua Eastwood
Numerade Educator
10:22

Problem 56

Group Activity Ship's Propeller The propellers of the Amazon Paradise have a radius of 1.2 $\mathrm{m}$ . At full throttle the propellers turn at 135 $\mathrm{rpm}$ .
(a) What is the angular speed of a propeller blade in radians per second?
(b) What is the linear speed of the tip of the propeller blade in meters per second?
(c) What is the linear speed (in meters per second) of a point on a blade halfway between the center of the propeller and the tip of the blade?

Eric Tung
Eric Tung
Numerade Educator
02:36

Problem 57

True or False If horse $A$ is twice as far as horse $B$ from the center of a merry-go-round, then horse $A$ travels twice as fast as horse $B .$ Justify your answer.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:37

Problem 58

True or False The radian measure of all three angles in a triangle can be integers. Justify your answer.

Eric Tung
Eric Tung
Numerade Educator
00:33

Problem 59

You may use a graphing calculator when answering these questions.
Multiple Choice What is the radian measure of an angle of $x$ degrees?
$$
\begin{array}{ll}{\text { (A) } \pi x} & {\text { (B) } x / 180} \\ {\text { (C) } \pi x / 180} & {\text { (D) } 180 x / \pi} \\ {\text { (E) } 180 / x \pi} \end{array}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
01:47

Problem 60

You may use a graphing calculator when answering these questions.
Multiple Choice If the perimeter of a sector is 4 times its radius, then the radian measure of the central angle of the sector is
$$
\begin{array}{ll}{(\mathbf{A}) 2} & {\text { (B) } 4} \\ {\text { (C) } 2 / \pi .} & {\text { (D) } 4 / \pi}\end{array}
$$
$$
\text {(E) impossible to determine without knowing the radius.}
$$

Eric Tung
Eric Tung
Numerade Educator
02:26

Problem 61

You may use a graphing calculator when answering these questions.
Multiple Choice A bicycle with 26 -inch-diameter wheels is traveling at 10 miles per hour. To the nearest whole number, how many revolutions does each wheel make per minute?
$$
\begin{array}{ll}{\text { (A) } 54} & {\text { (B) } 129} \\ {\text { (C) } 259} & {\text { (D) } 406} \\ {\text { (E) } 646} \end{array}
$$

Joshua Eastwood
Joshua Eastwood
Numerade Educator
01:17

Problem 62

You may use a graphing calculator when answering these questions.
Multiple Choice A central angle in a circle of radius $r$ has a measure of $\theta$ radians. If the same central angle were drawn in a circle of radius $2 r,$ its radian measure would be
$$
\begin{array}{ll}{\text { (A) } \frac{\theta}{2}} & {\text { (B) } \frac{\theta}{2 r}} \\ {\text { (C) } \theta} & {\text { (D) } 2 \theta} \\ {\text { (E) } 2 r \theta}} \end{array}
$$

Eric Tung
Eric Tung
Numerade Educator
02:11

Problem 63

In Exercises $63-66,$ find the difference in longitude between the given cities.
Atlanta and San Francisco

Joshua Eastwood
Joshua Eastwood
Numerade Educator
00:51

Problem 64

In Exercises $63-66,$ find the difference in longitude between the given cities.
New York and San Diego

Eric Tung
Eric Tung
Numerade Educator
01:46

Problem 65

In Exercises $63-66,$ find the difference in longitude between the given cities.
Minneapolis and Chicago

Joshua Eastwood
Joshua Eastwood
Numerade Educator
00:43

Problem 66

In Exercises $63-66,$ find the difference in longitude between the given cities.
Miami and Seattle

Eric Tung
Eric Tung
Numerade Educator
03:18

Problem 67

In Exercises $67-70$ , assume that the two cities have the same longitude (that is, assume that one is directly north of the other), and find the distance between them in nautical miles.
San Diego and Los Angeles

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:58

Problem 68

In Exercises $67-70$ , assume that the two cities have the same longitude (that is, assume that one is directly north of the other), and find the distance between them in nautical miles.
Seattle and San Francisco

Eric Tung
Eric Tung
Numerade Educator
03:46

Problem 69

In Exercises $67-70$ , assume that the two cities have the same longitude (that is, assume that one is directly north of the other), and find the distance between them in nautical miles.
New Orleans and Minneapolis

Joshua Eastwood
Joshua Eastwood
Numerade Educator
03:24

Problem 70

In Exercises $67-70$ , assume that the two cities have the same longitude (that is, assume that one is directly north of the other), and find the distance between them in nautical miles.
Detroit and Atlanta

Eric Tung
Eric Tung
Numerade Educator
01:48

Problem 71

Group Activity Area of a Sector A sector of $a$ circle (shaded in the figure) is a region bounded by a central angle of a circle and its intercepted arc. Use the fact that the areas of sectors are proportional to their central angles to prove that
$$
A=\frac{1}{2} r^{2} \theta
$$
where $r$ is the radius and $\theta$ is in radians.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
02:18

Problem 72

Area of a Seetor Use the formula $A=(1 / 2) r^{2} \theta$ to determine the area of the sector with given radius $r$ and central angle $\theta$ .
(a) $r=5.9 \mathrm{ft}, \theta=\pi / 5$
(b) $r=1.6 \mathrm{km}, \theta=3.7$

Eric Tung
Eric Tung
Numerade Educator
01:45

Problem 73

Navigation Control tower $A$ is 60 miles east of control tower $B$ . At a certain time an airplane is on bearings of $340^{\circ}$ from tower $A$ and $37^{\circ}$ from tower $B .$ Use a drawing to model the exact location of the airplane.

Joshua Eastwood
Joshua Eastwood
Numerade Educator
04:58

Problem 74

Bicycle Racing Ben Scheltz's bike wheels are 28 inches in diameter, and for high gear the pedal sprocket is 9 inches in diameter and the wheel sprocket is 3 inches in diameter. Find the angular speed in radians per second of the wheel and of both sprockets when Ben reaches his peak racing speed of 66 ft/sec in high gear.

Eric Tung
Eric Tung
Numerade Educator