Precalculus with Limits

Ron Larson

Chapter 4

Trigonometry - all with Video Answers

Educators

+ 22 more educators

Section 1

Radian and Degree Measure

Problem 1

Fill in the blanks.
Two angles that have the same initial and terminal sides are _______ .

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 2

Fill in the blanks.
One _______ is the measure of a central angle that intercepts an arc equal to the radius of the circle.

K Joseph

Numerade Educator

Problem 3

Fill in the blanks.
Two positive angles that have a sum of $\pi / 2$ are _______ angles, whereas two positive angles that have a sum of $\pi$ are _______ angles.

Bobby Barnes

University of North Texas

Problem 4

Fill in the blanks.
The angle measure that is equivalent to a rotation of a complete revolution about an angle's vertex is one _______ .

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 5

Fill in the blanks.
The ______ speed of a particle is the ratio of the arc length to the time traveled, and the the ______ speed of a particle is the ratio of the central angle to the time traveled.

K Joseph

Numerade Educator

Problem 6

Fill in the blanks.
The area $A$ of a sector of a circle with radius $r$ and central angle $\theta,$ where $\theta$ is measured in radians, is given by the formula ______ .

K Joseph

Numerade Educator

Problem 7

Estimate the angle to the nearest one-half radian.

Charles Carter

Numerade Educator

Problem 8

Estimate the angle to the nearest one-half radian.

Shazia Naz

Numerade Educator

Problem 9

Estimate the angle to the nearest one-half radian.

Carson Merrill

Numerade Educator

Problem 10

Estimate the angle to the nearest one-half radian.

Rowan Ahmed

Numerade Educator

Problem 11

Determine the quadrant in which each angle lies.
(a) $\frac{\pi}{4} \quad$ (b) $\frac{5 \pi}{4}$

K Joseph

Numerade Educator

Problem 12

Determine the quadrant in which each angle lies.
$(\mathrm{a})-\frac{\pi}{6} \quad(\mathrm{b})-\frac{11 \pi}{9}$

Anh Nguyen

Numerade Educator

Problem 13

Sketch each angle in standard position.
(a) $\frac{\pi}{3} \quad(\mathrm{b})-\frac{2 \pi}{3}$

K Joseph

Numerade Educator

Problem 14

Sketch each angle in standard position.
(a) $\frac{5 \pi}{2} \quad$ (b) 4

K Joseph

Numerade Educator

Problem 15

Determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians.
(a) $\frac{\pi}{6} \quad$ (b) $\frac{7 \pi}{6}$

Sheryl Ezze

Numerade Educator

Problem 16

Determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians.
(a) $\frac{2 \pi}{3} \quad(\mathrm{b})-\frac{9 \pi}{4}$

K Joseph

Numerade Educator

Problem 17

Find (if possible) the complement and the supplement of each angle.
(a) $\frac{\pi}{3} \quad$ (b) $\frac{\pi}{4}$

K Joseph

Numerade Educator

Problem 18

Find (if possible) the complement and the supplement of each angle.
(a) $\frac{\pi}{12} \quad$ (b) $\frac{11 \pi}{12}$

K Joseph

Numerade Educator

Problem 19

Find (if possible) the complement and the supplement of each angle.
(a) 1 (b) 2

K Joseph

Numerade Educator

Problem 20

Find (if possible) the complement and the supplement of each angle.
(a) 3 (b) 1.5

K Joseph

Numerade Educator

Problem 21

The number of degrees in the angle.

K Joseph

Numerade Educator

Problem 22

The number of degrees in the angle.

K Joseph

Numerade Educator

Problem 23

Estimate the number of degrees in the angle.

K Joseph

Numerade Educator

Problem 24

The number of degrees in the angle.

K Joseph

Numerade Educator

Problem 25

Determine the quadrant in which each angle lies.
$\begin{array}{ll}{\text { (a) } 130^{\circ}} & {\text { (b) } 8.3^{\circ}}\end{array}$

K Joseph

Numerade Educator

Problem 26

Determine the quadrant in which each angle lies.
(a) $-132^{\circ} 50^{\prime}$ (b) $-3.4^{\circ}$

K Joseph

Numerade Educator

Problem 27

Sketch each angle in standard position.
(a) $270^{\circ} \quad$ (b) $120^{\circ}$

Derrick Hanson

Numerade Educator

Problem 28

Sketch each angle in standard position.
(a) $-135^{\circ} \quad$ (b) $-750^{\circ}$

ZC

Zara Cano Molina

Numerade Educator

Problem 29

determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees.
(a) $45^{\circ} \quad$ (b) $-36^{\circ}$

Bryanna Ehly

Numerade Educator

Problem 30

determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees.
(a) $120^{\circ} \quad$ (b) $-420^{\circ}$

James Kiss

Numerade Educator

Problem 31

Find (if possible) the complement and the supplement of each angle.
(a) $18^{\circ} \quad$ (b) $85^{\circ}$

K Joseph

Numerade Educator

Problem 32

Find (if possible) the complement and the supplement of each angle.
(a) $46^{\circ} \quad$ (b) $93^{\circ}$

K Joseph

Numerade Educator

Problem 33

Find (if possible) the complement and the supplement of each angle.
(a) $150^{\circ} \quad$ (b) $79^{\circ}$

K Joseph

Numerade Educator

Problem 34

Find (if possible) the complement and the supplement of each angle.
(a) $130^{\circ} \quad$ (b) $170^{\circ}$

K Joseph

Numerade Educator

Problem 35

Rewrite each angle in radian measure as a multiple of $\pi .$ (Do not use a calculator.)
(a) $120^{\circ} \quad$ (b) $-20^{\circ}$

Madi Sousa

Numerade Educator

Problem 36

Rewrite each angle in radian measure as a multiple of $\pi .$ (Do not use a calculator.)
$\begin{array}{ll}{\text { (a) }-60^{\circ}} & {\text { (b) } 144^{\circ}}\end{array}$

K Joseph

Numerade Educator

Problem 37

Rewrite each angle in degree measure. (Do not use a calculator.)
(a) $\frac{3 \pi}{2} \quad$ (b) $\frac{7 \pi}{6}$

AG

Ankit Gupta

Numerade Educator

Problem 38

Rewrite each angle in degree measure. (Do not use a calculator.)
$(\mathrm{a})-\frac{7 \pi}{12} \quad$ (b) $\frac{5 \pi}{4}$

K Joseph

Numerade Educator

Problem 39

Convert the angle measure from degrees to radians. Round to three decimal places.
$45^{\circ}$

K Joseph

Numerade Educator

Problem 40

Convert the angle measure from degrees to radians. Round to three decimal places.
$-48.27^{\circ}$

K Joseph

Numerade Educator

Problem 41

Convert the angle measure from degrees to radians. Round to three decimal places.
$0.54^{\circ}$

K Joseph

Numerade Educator

Problem 42

Convert the angle measure from degrees to radians. Round to three decimal places.
$345^{\circ}$

Jessica Mandel

Numerade Educator

Problem 43

Convert the angle measure from radians to degrees. Round to three decimal places.
$\frac{5 \pi}{11}$

K Joseph

Numerade Educator

Problem 44

Convert the angle measure from radians to degrees. Round to three decimal places.
$\frac{15 \pi}{8}$

K Joseph

Numerade Educator

Problem 45

Convert the angle measure from radians to degrees. Round to three decimal places.
$-4.2 \pi$

Lv

Leandra Van Der Merwe

Numerade Educator

Problem 46

Convert the angle measure from radians to degrees. Round to three decimal places.
$-0.57$

K Joseph

Numerade Educator

Problem 47

Convert each angle measure to decimal degree form without using a calculator. Then check your
answers using a calculator.
(a) $54^{\circ} 45^{\prime}$

K Joseph

Numerade Educator

Problem 48

Convert each angle measure to decimal degree form without using a calculator. Then check your
answers using a calculator.
(a) $-135^{\circ} 36 "$

K Joseph

Numerade Educator

Problem 49

Convert each angle measure to degrees, minutes, and seconds without using a calculator. Then check your answers using a calculator.
(a) $240.6^{\circ} \quad$ (b) $-145.8^{\circ}$

K Joseph

Numerade Educator

Problem 50

Convert each angle measure to degrees, minutes, and seconds without using a calculator. Then check your answers using a calculator.
(a) $-345.12^{\circ} \quad$ (b) $-3.58^{\circ}$

K Joseph

Numerade Educator

Problem 51

Determine two coterminal angles fone positive and one negative) for each angle. Give your answers in degrees.
(a) $\theta=240^{\circ}$
(b) $\theta=-180^{\circ}$

Erika Bustos

Numerade Educator

Problem 52

Find the length of the arc on a circle of radius $r$ intercepted by a central angle $\theta .$
$r=3$ meters, $\theta=150^{\circ}$

K Joseph

Numerade Educator

Problem 53

Find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$ .
$r=80$ kilometers, $s=150$ kilometers

K Joseph

Numerade Educator

Problem 54

Find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$ .
$r=14$ feet $, s=8$ feet

K Joseph

Numerade Educator

Problem 55

Use the given arc length and radius to find the angle $\theta$ (in radians).

Harmender Singh Yadav

Harmender Singh Yadav

Numerade Educator

Problem 56

Finding an Angle In Exercises 55 and $56,$ use the given arc length and radius to find the angle $\theta$ (in radians).

DB

David Braunlich

Numerade Educator

Problem 57

Find the area of the sector of a circle of radius $r$ and central angle $\theta .$
$r=12$ millimeters, $\theta=\frac{\pi}{4}$

K Joseph

Numerade Educator

Problem 58

Find the area of the sector of a circle of radius $r$ and central angle $\theta .$
$r=2.5$ feet $, \theta=225^{\circ}$

K Joseph

Numerade Educator

Problem 59

Between Dallas, Texas, whose latitude is $32^{\circ} 47^{\prime} 39^{\prime \prime} \mathrm{N}$
and Omaha, Nebraska, whose latitude is $41^{\circ} 15^{\prime} 50^{\prime \prime} \mathrm{N}$
Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (Omaha is due north of Dallas).

Linda Hand

Numerade Educator

Problem 60

Assuming that Earth is a sphere of radius 6378 kilometers, what is the difference in the latitudes of Lynchburg, Virginia, and Myrtle Beach, South Carolina, where Lynchburg is about 400 kilometers
due north of Myrtle Beach?

Sheryl Ezze

Numerade Educator

Problem 61

The pointer on a voltmeter is 6 centimeters in length (see figure). Find the number of degrees through which the pointer rotates when it moves 2.5 centimeters on the scale.

Stephen Hobbs

Numerade Educator

Problem 62

A satellite in a circular orbit 1250 kilometers above Earth makes one complete revolution every 110 minutes. Assuming that Earth is a sphere of radius 6378 kilometers, what is the linear speed (in kilometers per minute) of the satellite?

Sheryl Ezze

Numerade Educator

Problem 63

The circular blade on a saw rotates at 5000 revolutions per minute.
(a) Find the angular speed of the blade in radians per minute.
(b) The blade has a diameter of 7$\frac{1}{4}$ inches. Find the linear speed of a blade tip.

Linda Hand

Numerade Educator

Problem 64

A carousel with a 50 -foot diameter makes 4 revolutions per minute.
(a) Find the angular speed of the carousel in radians per minute.
(b) Find the linear speed (in feet per minute) of the platform rim of the carousel.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 65

A DVD is approximately 12 centimeters in diameter. The drive motor of the DVD player rotates between 200 and 500 revolutions per minute, depending on what track is being read. (a) Find an interval for the angular speed of the DVD as it rotates.

Suzanne W.

Numerade Educator

Problem 66

A car is moving at a rate of 65 miles per hour, and the diameter of its wheels is 2 feet.
(a) Find the number of revolutions per minute the wheels are rotating.
(b) Find the angular speed of the wheels in radians per minute.

Khushbu Rani

Numerade Educator

Problem 67

A computerized spin balance machine rotates a 25 -inch-diameter tire at 480 revolutions per minute.
(a) Find the road speed (in miles per hour) at which the tire is being balanced.
(b) At what rate should the spin balance machine be set so that the tire is being tested for 55 miles per hour?

Ahmed Kamel

Numerade Educator

Problem 68

The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are 4 inches, 2 inches, and 14 inches, respectively. A cyclist is pedaling at a rate of 1 revolution per second
(a) Find the speed of the bicycle in feet per second and miles per hour.
(b) Use your result from part (a) to write a function for the distance $d$ (in miles) a cyclist travels in terms of the number $n$ of revolutions of the pedal sprocket.

AG

Ankit Gupta

Numerade Educator

Problem 69

A sprinkler on a golf green sprays water over a distance of 15 meters and rotates through an angle of
$140^{\circ} .$ Draw a diagram that shows the region that the sprinkler can irrigate. Find the area of the region.

Stephen Hobbs

Numerade Educator

Problem 70

A car's rear windshield wiper rotates $125^{\circ} .$ The total length of the wiper mechanism is 25 inches and wipes the windshield over a distance of 14 inches. Find the area covered by the wiper.

Anas Venkitta

Numerade Educator

Problem 71

Determine whether the statement is true or false. Justify your answer.
A measurement of 4 radians corresponds to two complete revolutions from the initial side to the terminal side of an angle.

K Joseph

Numerade Educator

Problem 72

Determine whether the statement is true or false. Justify your answer.
The difference between the measures of two coterminal angles is always a multiple of $360^{\circ}$ when expressed in degrees and is always a multiple of 2$\pi$ radians when expressed in radians.

K Joseph

Numerade Educator

Problem 73

Determine whether the statement is true or false. Justify your answer.
An angle that measures $-1260^{\circ}$ lies in Quadrant III.

AG

Ankit Gupta

Numerade Educator

Problem 74

Determine which angles in the figure are coterminal angles with angle $A .$ Explain your reasoning.

AG

Ankit Gupta

Numerade Educator

Problem 75

$A$ fan motor turns at a given angular speed. How does the speed of the tips of the blades change when a fan of greater diameter is on the motor? Explain.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator

Problem 76

Is a degree or a radian the greater unit of measure? Explain.

Sarah Vo

Numerade Educator

Problem 77

When the radius of a circle increases and the magnitude of a central angle is constant, how does the
length of the intercepted arc change? Explain your reasoning.

K Joseph

Numerade Educator

Problem 78

Prove that the area of a circular sector of radius $r$ with central angle $\theta$ is
$$A=\frac{1}{2} \theta r^{2}$$
where $\theta$ is measured in radians.

Suman Saurav Thakur

Suman Saurav Thakur

Numerade Educator