An ________ is determined by rotating a ray about its endpoint.

Heather Z.

Numerade Educator

Two angles that have the same initial and terminal sides are ________.

Heather Z.

Numerade Educator

One ________ is the measure of a central angle that intercepts an arc equal to the radius of the circle.

Heather Z.

Numerade Educator

Angles that measure between $0$ and $\pi/2$ are ________ angles, and angles that measure between $\pi/2$ and $\pi$ are ________ angles.

Heather Z.

Numerade Educator

Two positive angles that have a sum of $\pi/2$ are ________ angles, whereas two positive angles that have a sum of $\pi$ are ________ angles.

Heather Z.

Numerade Educator

The angle measure that is equivalent to a rotation of $\frac{1}{360}$ of a complete revolution about an angle's vertex is one ________.

Heather Z.

Numerade Educator

The ________ speed of a particle is the ratio of arc length to time traveled, and the ________ speed of a particle is the ratio of central angle to time traveled.

Heather Z.

Numerade Educator

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

Heather Z.

Numerade Educator

In Exercises 11-16, estimate the angle to the nearest one-half radian.

Heather Z.

Numerade Educator

In Exercises 11-16, estimate the angle to the nearest one-half radian.

Heather Z.

Numerade Educator

In Exercises 11-16, estimate the angle to the nearest one-half radian.

Heather Z.

Numerade Educator

In Exercises 11-16, estimate the angle to the nearest one-half radian.

Heather Z.

Numerade Educator

In Exercises 11-16, estimate the angle to the nearest one-half radian.

Heather Z.

Numerade Educator

In Exercises 11-16, estimate the angle to the nearest one-half radian.

Heather Z.

Numerade Educator

In Exercises 17-22, determine the quadrant in which each angle lies. (The angle measure is given in radians.)

(a) $\frac{\pi}{4}$

(b) $\frac{5\pi}{4}$

Heather Z.

Numerade Educator

In Exercises 17-22, determine the quadrant in which each angle lies. (The angle measure is given in radians.)

(a) $\frac{11\pi}{8}$

(b) $\frac{9\pi}{8}$

Heather Z.

Numerade Educator

In Exercises 17-22, determine the quadrant in which each angle lies. (The angle measure is given in radians.)

(a) $-\frac{\pi}{6}$

(b) $-\frac{\pi}{3}$

Heather Z.

Numerade Educator

(a) $-\frac{5\pi}{6}$

(b) $-\frac{11\pi}{9}$

Heather Z.

Numerade Educator

(a) $3.5$

(b) $2.25$

Heather Z.

Numerade Educator

(a) $6.02$

(b) $-4.25$

Heather Z.

Numerade Educator

In Exercises 23-26, sketch each angle in standard position.

(a) $\frac{\pi}{3}$

(b) $-\frac{2\pi}{3}$

Heather Z.

Numerade Educator

In Exercises 23-26, sketch each angle in standard position.

(a) $-\frac{7\pi}{4}$

(b) $\frac{5\pi}{2}$

Heather Z.

Numerade Educator

In Exercises 23-26, sketch each angle in standard position.

(a) $\frac{11\pi}{6}$

(b) $-3$

Heather Z.

Numerade Educator

In Exercises 23-26, sketch each angle in standard position.

(a) $4$

(b) $7\pi$

Heather Z.

Numerade Educator

In Exercises 27-30, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians.

(a) $\theta = \frac{\pi}{6}$

(b) $\theta = \frac{5\pi}{6}$

Heather Z.

Numerade Educator

In Exercises 27-30, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians.

(a) $\theta = \frac{7\pi}{6}$

(b) $\theta = -\frac{11\pi}{6}$

Heather Z.

Numerade Educator

In Exercises 27-30, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in radians.

(a) $\theta = \frac{2\pi}{3}$

(b) $\theta = \frac{\pi}{12}$

Heather Z.

Numerade Educator

(a) $\theta = -\frac{9\pi}{4}$

(b) $\theta = -\frac{2\pi}{15}$

Heather Z.

Numerade Educator

In Exercises 31-34, find (if possible) the complement and supplement of each angle.

(a) $\pi /3$

(b) $\pi /4$

Heather Z.

Numerade Educator

In Exercises 31-34, find (if possible) the complement and supplement of each angle.

(a) $\pi /12$

(b) $11\pi /12$

Heather Z.

Numerade Educator

In Exercises 31-34, find (if possible) the complement and supplement of each angle.

(a) $1$

(b) $2$

Heather Z.

Numerade Educator

In Exercises 31-34, find (if possible) the complement and supplement of each angle.

(a) $3$

(b) $1.5$

Heather Z.

Numerade Educator

In Exercises 35-40, estimate the number of degrees in the angle. Use a protractor to check your answer.

Heather Z.

Numerade Educator

Heather Z.

Numerade Educator

Heather Z.

Numerade Educator

Heather Z.

Numerade Educator

Heather Z.

Numerade Educator

Heather Z.

Numerade Educator

In Exercises 41-44, determine the quadrant in which each angle lies.

(a) $130^{\circ}$

(b) $285^{\circ}$

Heather Z.

Numerade Educator

In Exercises 41-44, determine the quadrant in which each angle lies.

(a) $8.3^{\circ}$

(b) $257^{\circ} 30'$

Heather Z.

Numerade Educator

In Exercises 41-44, determine the quadrant in which each angle lies.

(a) $-132^{\circ} 50'$

(b) $-336^{\circ}$

Heather Z.

Numerade Educator

In Exercises 41-44, determine the quadrant in which each angle lies.

(a) $-260^{\circ}$

(b) $-3.4^{\circ}$

Heather Z.

Numerade Educator

In Exercises 45-48, sketch each angle in standard position.

(a) $90^{\circ}$

(b) $180^{\circ}$

Heather Z.

Numerade Educator

In Exercises 45-48, sketch each angle in standard position.

(a) $270^{\circ}$

(b) $120^{\circ}$

Heather Z.

Numerade Educator

In Exercises 45-48, sketch each angle in standard position.

(a) $-30^{\circ}$

(b) $-135^{\circ}$

Heather Z.

Numerade Educator

In Exercises 45-48, sketch each angle in standard position.

(a) $-750^{\circ}$

(b) $-600^{\circ}$

Heather Z.

Numerade Educator

In Exercises 49-52, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees.

(a) $\theta = 45^{\circ}$

(b) $\theta = -36^{\circ}$

Heather Z.

Numerade Educator

In Exercises 49-52, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees.

(a) $\theta = 120^{\circ}$

(b) $\theta = -420^{\circ}$

Heather Z.

Numerade Educator

In Exercises 49-52, determine two coterminal angles (one positive and one negative) for each angle. Give your answers in degrees.

(a) $\theta = 240^{\circ}$

(b) $\theta = -180^{\circ}$

Heather Z.

Numerade Educator

(a) $\theta = -390^{\circ}$

(b) $\theta = 230^{\circ}$

Heather Z.

Numerade Educator

In Exercises 53-56, find (if possible) the complement and supplement of each angle.

(a) $18^{\circ}$

(b) $85^{\circ}$

Heather Z.

Numerade Educator

In Exercises 53-56, find (if possible) the complement and supplement of each angle.

(a) $46^{\circ}$

(b) $93^{\circ}$

Heather Z.

Numerade Educator

In Exercises 53-56, find (if possible) the complement and supplement of each angle.

(a) $150^{\circ}$

(b) $79^{\circ}$

Heather Z.

Numerade Educator

In Exercises 53-56, find (if possible) the complement and supplement of each angle.

(a) $130^{\circ}$

(b) $170^{\circ}$

Heather Z.

Numerade Educator

In Exercises 57-60, rewrite each angle in radian measure as a multiple of $\pi$. (Do not use a calculator.)

(a) $30^{\circ}$

(b) $45^{\circ}$

Heather Z.

Numerade Educator

In Exercises 57-60, rewrite each angle in radian measure as a multiple of $\pi$. (Do not use a calculator.)

(a) $315^{\circ}$

(b) $120^{\circ}$

Heather Z.

Numerade Educator

In Exercises 57-60, rewrite each angle in radian measure as a multiple of $\pi$. (Do not use a calculator.)

(a) $-20^{\circ}$

(b) $-60^{\circ}$

Heather Z.

Numerade Educator

Rewrite each angle in radian measure as a multiple of $\pi$.

(a) $-270^{\circ}$

(b) $144^{\circ}$

Heather Z.

Numerade Educator

In Exercises 61-64, rewrite each angle in degree measure.(Do not use a calculator.)

(a) $\frac{3\pi}{2}$

(b) $\frac{7\pi}{2}$

Heather Z.

Numerade Educator

In Exercises 61-64, rewrite each angle in degree measure.(Do not use a calculator.)

(a) $-\frac{7\pi}{12}$

(b) $\frac{\pi}{9}$

Heather Z.

Numerade Educator

In Exercises 61-64, rewrite each angle in degree measure.(Do not use a calculator.)

(a) $\frac{5\pi}{4}$

(b) $-\frac{7\pi}{3}$

Heather Z.

Numerade Educator

In Exercises 61-64, rewrite each angle in degree measure.(Do not use a calculator.)

(a) $\frac{11\pi}{6}$

(b) $\frac{34\pi}{15}$

Heather Z.

Numerade Educator

In Exercises 65-72, convert the angle measure from degrees to radians. Round to three decimal places.

$45^{\circ}$

Heather Z.

Numerade Educator

In Exercises 65-72, convert the angle measure from degrees to radians. Round to three decimal places.

$87.4^{\circ}$

Heather Z.

Numerade Educator

In Exercises 65-72, convert the angle measure from degrees to radians. Round to three decimal places.

$-216.35^{\circ}$

Heather Z.

Numerade Educator

$-48.27^{\circ}$

Heather Z.

Numerade Educator

$532^{\circ}$

Heather Z.

Numerade Educator

$345^{\circ}$

Heather Z.

Numerade Educator

$-0.83^{\circ}$

Heather Z.

Numerade Educator

$0.54^{\circ}$

Heather Z.

Numerade Educator

In Exercises 73-80, convert the angle measure from radians to degrees. Round to three decimal places.

$\pi/7$

Heather Z.

Numerade Educator

In Exercises 73-80, convert the angle measure from radians to degrees. Round to three decimal places.

$5\pi/11$

Heather Z.

Numerade Educator

In Exercises 73-80, convert the angle measure from radians to degrees. Round to three decimal places.

$15\pi/8$

Heather Z.

Numerade Educator

$13\pi/2$

Heather Z.

Numerade Educator

$-4.2\pi$

Heather Z.

Numerade Educator

convert the angle measure from radians to degrees. Round to three decimal places. $4.8 \pi$

Heather Z.

Numerade Educator

$-2$

Heather Z.

Numerade Educator

$-0.57$

Heather Z.

Numerade Educator

In Exercises 81-84, convert each angle measure to decimal degree form without using a calculator. Then check your answers using a calculator.

(a) $54^{\circ} 45'$

(b) $-128^{\circ} 30'$

Heather Z.

Numerade Educator

In Exercises 81-84, convert each angle measure to decimal degree form without using a calculator. Then check your answers using a calculator.

(a) $245^{\circ} 10'$

(b) $2^{\circ} 12'$

Heather Z.

Numerade Educator

In Exercises 81-84, convert each angle measure to decimal degree form without using a calculator. Then check your answers using a calculator.

(a) $85^{\circ} 18' 30''$

(b) $330^{\circ} 25''$

Heather Z.

Numerade Educator

(a) $-135^{\circ} 36''$

(b) $-408^{\circ} 16' 20''$

Heather Z.

Numerade Educator

In Exercises 85-88, convert each angle measure to degrees,minutes, and seconds without using a calculator. Then check your answers using a calculator.

(a) $240.6^{\circ}$

(b) $-145.8^{\circ}$

Heather Z.

Numerade Educator

In Exercises 85-88, convert each angle measure to degrees,minutes, and seconds without using a calculator. Then check your answers using a calculator.

(a) $-345.12^{\circ}$

(b) $0.45^{\circ}$

Heather Z.

Numerade Educator

In Exercises 85-88, convert each angle measure to degrees,minutes, and seconds without using a calculator. Then check your answers using a calculator.

(a) $2.5^{\circ}$

(b) $-3.58^{\circ}$

Heather Z.

Numerade Educator

(a) $-0.36^{\circ}$

(b) $0.79^{\circ}$

Heather Z.

Numerade Educator

In Exercises 89-92, find the length of the arc on a circle of radius $r$ intercepted by a central angle $theta$.

$Radius$ $r$

15 inches

$Central$ $Angle$ $\theta$

$120^{\circ}$

Heather Z.

Numerade Educator

In Exercises 89-92, find the length of the arc on a circle of radius $r$ intercepted by a central angle $theta$.

$Radius$ $r$

9 feet

$Central$ $Angle$ $\theta$

$60^{\circ}$

Heather Z.

Numerade Educator

In Exercises 89-92, find the length of the arc on a circle of radius $r$ intercepted by a central angle $theta$.

$Radius$ $r$

3 meter

$Central$ $Angle$ $\theta$

$150^{\circ}$

Heather Z.

Numerade Educator

$Radius$ $r$

20 centimeters

$Central$ $Angle$ $\theta$

$45^{\circ}$

Heather Z.

Numerade Educator

In Exercises 93-96, find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$.

$Radius$ $r$

4 inches

$Arc$ $Length$ $s$

18 inches

Heather Z.

Numerade Educator

In Exercises 93-96, find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$.

$Radius$ $r$

14 feet

$Arc$ $Length$ $s$

8 feet

Heather Z.

Numerade Educator

In Exercises 93-96, find the radian measure of the central angle of a circle of radius $r$ that intercepts an arc of length $s$.

$Radius$ $r$

25 centimeters

$Arc$ $Length$ $s$

10.5 centimeters

Heather Z.

Numerade Educator

$Radius$ $r$

80 kilometers

$Arc$ $Length$ $s$

150 kilometers

Heather Z.

Numerade Educator

In Exercises 97-100, use the given arc length and radius to find the angle $\theta$ (in radians).

Heather Z.

Numerade Educator

In Exercises 97-100, use the given arc length and radius to find the angle $\theta$ (in radians).

Heather Z.

Numerade Educator

In Exercises 97-100, use the given arc length and radius to find the angle $\theta$ (in radians).

Heather Z.

Numerade Educator

In Exercises 97-100, use the given arc length and radius to find the angle $\theta$ (in radians).

Heather Z.

Numerade Educator

In Exercises 101-104, find the area of the sector of the circle with radius $r$ and central angle $\theta$.

$Radius$ $r$

6 inches

$Central$ $Angle$ $\theta$

$\pi/3$

Heather Z.

Numerade Educator

In Exercises 101-104, find the area of the sector of the circle with radius $r$ and central angle $\theta$.

$Radius$ $r$

12 millimeters

$Central$ $Angle$ $\theta$

$\pi/4$

Heather Z.

Numerade Educator

In Exercises 101-104, find the area of the sector of the circle with radius $r$ and central angle $\theta$.

$Radius$ $r$

2.5 feet

$Central$ $Angle$ $\theta$

225$^\circ$

Heather Z.

Numerade Educator

$Radius$ $r$

1.4 miles

$Central$ $Angle$ $\theta$

330$^\circ$

Heather Z.

Numerade Educator

DISTANCE BETWEEN CITIES In Exercises 105 and 106, find the distance between the cities. Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (one city is due north of the other).

$City$

Dallas, Texas

$Latitude$

$32^{\circ} 47' 39'' N$

$City$

Omaha, Nebraska

$Latitude$

$41^{\circ} 15' 50'' N$

Heather Z.

Numerade Educator

DISTANCE BETWEEN CITIES In Exercises 105 and 106, find the distance between the cities. Assume that Earth is a sphere of radius 4000 miles and that the cities are on the same longitude (one city is due north of the other).

$City$

San Francisco, California

$Latitude$

$37^{\circ} 47' 36'' N$

$City$

Seattle, Washington

$Latitude$

$47^{\circ} 37' 18'' N$

Heather Z.

Numerade Educator

DIFFERENCE IN LATITUDES Assuming that Earth is a sphere of radius 6378 kilometers, what is the difference in the latitudes of Syracuse, New York and Annapolis, Maryland, where Syracuse is about 450 kilometers due north of Annapolis?

Heather Z.

Numerade Educator

DIFFERENCE IN LATITUDES 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?

Heather Z.

Numerade Educator

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

Heather Z.

Numerade Educator

ELECTRIC HOIST An electric hoist is being used to lift a beam (see figure). The diameter of the drum on the hoist is 10 inches, and the beam must be raised 2 feet. Find the number of degrees through which the drum must rotate.

Heather Z.

Numerade Educator

LINEAR AND ANGULAR SPEEDS A circular power saw has a $7\frac{1}{4}$-inch-diameter blade that rotates at 5000 revolutions per minute.

(a) Find the angular speed of the saw blade in radians per minute.

(b) Find the linear speed (in feet per minute) of one of the 24 cutting teeth as they contact the wood being cut.

Heather Z.

Numerade Educator

LINEAR AND ANGULAR SPEEDS 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.

Heather Z.

Numerade Educator

LINEAR AND ANGULAR SPEEDS The diameter of a DVD is approximately 12 centimeters. The drive motor of the DVD player is controlled to rotate precisely between 200 and 500 revolutions per minute, depending on what track is being read.

(a) Find an interval for the angular speed of a DVD as it rotates.

(b) Find an interval for the linear speed of a point on the outermost track as the DVD rotates.

Heather Z.

Numerade Educator

ANGULAR SPEED A two-inch-diameter pulley on an electric motor that runs at 1700 revolutions per minute is connected by a belt to a four-inch-diameter pulley on a saw arbor.

(a) Find the angular speed (in radians per minute) of each pulley.

(b) Find the revolutions per minute of the saw.

Heather Z.

Numerade Educator

ANGULAR SPEED 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.

Heather Z.

Numerade Educator

ANGULAR SPEED 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?

Heather Z.

Numerade Educator

AREA A sprinkler on a golf green is set to spray water over a distance of 15 meters and to rotate through an angle of $140^{\circ}$. Draw a diagram that shows the region that can be irrigated with the sprinkler. Find the area of the region.

Heather Z.

Numerade Educator

AREA 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.

Heather Z.

Numerade Educator

SPEED OF A BICYCLE 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.

(c) Write a function for the distance $d$ (in miles) a cyclist travels in terms of the time $t$ (in seconds). Compare this function with the $t$ function from part (b).

(d) Classify the types of functions you found in parts(b) and (c). Explain your reasoning.

Heather Z.

Numerade Educator

CAPSTONE Write a short paper in your own words explaining the meaning of each of the following concepts to a classmate.

(a) an angle in standard position

(b) positive and negative angles

(c) coterminal angles

(d) angle measure in degrees and radians

(e) obtuse and acute angles

(f ) complementary and supplementary angles

Heather Z.

Numerade Educator

TRUE OR FALSE? In Exercises 121-123, 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.

Heather Z.

Numerade Educator

TRUE OR FALSE? In Exercises 121-123, 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 $360^{\circ}$ if expressed in degrees and is always a multiple of $2\pi$ radians if expressed in radians.

Heather Z.

Numerade Educator

TRUE OR FALSE? In Exercises 121-123, determine whether the statement is true or false. Justify your answer.

An angle that measures $ - 1260^{\circ}$ lies in Quadrant III.

Heather Z.

Numerade Educator

THINK ABOUT IT A fan motor turns at a given angular speed. How does the speed of the tips of the blades change if a fan of greater diameter is installed on the motor? Explain.

Heather Z.

Numerade Educator

THINK ABOUT IT Is a degree or a radian the larger unit of measure? Explain.

Heather Z.

Numerade Educator

WRITING If the radius of a circle is increasing and the magnitude of a central angle is held constant, how is the length of the intercepted arc changing? Explain your reasoning.

Heather Z.

Numerade Educator

PROOF 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.

Heather Z.

Numerade Educator