Radian Measure and the Unit Circle Approach

3

Radian Measure

and the Unit Circle

Approach

H

ow does an odometer or

speedometer on an automobile

work? The transmission counts how

many times the tires rotate (how many full revolutions take place) per second. A computer then calculates

how far the car has traveled in that second by multiplying the number of revolutions by the tire

circumference. Distance is given by the odometer, and the speedometer takes the distance per second

and converts to miles per hour (or km/h). Realize that the computer chip is programmed to the tire

designed for the vehicle. If a person were to change the tire size (smaller or larger than the original

specifications), then the odometer and speedometer would need to be adjusted.

Suppose you bought a Ford Expedition Eddie Bauer Edition, which comes standard with 17-inch rims

(corresponding to a tire with 25.7-inch diameter), and you decide to later upgrade these tires for 19-inch

rims (corresponding to a tire with 28.2-inch diameter). If the onboard computer is not adjusted, is the

actual speed faster or slower than the speedometer indicator?*

In this case, the speedometer would read 9.6% too low. For example, if your speedometer read 60 mph,

your actual speed would be 65.8 mph. In this chapter, you will see how the

angular speed

(rotations of

tires per second),

radius

(of the tires), and

linear speed

(speed of the automobile) are related.

C our tesy Fo rd Motor C ompany

I N T H I S C H A P T E R, you will learn a second way to measure angles using radians. You will convert between degrees and radians. You will calculate arc lengths, areas of circular sectors, and angular and linear speeds. Finally, the third definition of trigonometric functions using the unit circle approach will be given. You will work with the trigonometric functions in the context of a unit circle.

129 129

• Arc Length

• Area of a Circular

Sector

• Trigonometric

Functions and the

Unit Circle (Circular

Functions)

• Properties of Circular

Functions

• The Radian Measure

of an Angle

• Converting Between

Degrees and Radians

• Linear Speed

• Angular Speed

• Relationship

Between Linear and

Angular Speeds

3.1

Radian Measure

3.3

Linear and

Angular Speeds

3.4

Definition 3 of

Trigonometric

Functions: Unit

Circle Approach

3.2

Arc Length

and Area of a

Circular Sector

■

Convert between degrees and radians.

■

Calculate arc length and the area of a circular sector.

■

Relate angular and linear speeds.

■

Draw the unit circle and label the sine and cosine values for special angles

(in both degrees and radians).

R A D I A N M E A S U R E A N D T H E

U N I T C I R C L E A P P R OA C H

L E A R N I N G O B J E C T I V E S c03.qxd 8/22/11 7:07 PM Page 129

The Radian Measure of an Angle

In geometry and most everyday applications, angles are measured in degrees. However,

radian measure

is another way to measure angles. Using radian measure allows us to write

trigonometric functions as functions not only of angles but also of real numbers in general.

Recall that in Section 1.1 we defined one full rotation as an angle having measure

Now we think of the angle in the context of a circle. A central angle

is an angle that has

its vertex at the center of a circle.

When the intercepted arc’s length is equal to the radius, the measure of the central angle

is 1 radian. From geometry, we know that the ratio of the measures of two angles is equal

to the ratio of the lengths of the arcs subtended by those angles (along the same circle).

u

1

u

2

s

1

s

2

360°.

r ␪ = 1 radian r r

Note that both

s

and

r

are measured in units of length. When both are given in the same

units, the units cancel, giving the number of radians as a

dimensionless

(unitless) real

number.

C A U T I O N

To correctly calculate radians from the formula the radius and arc length must be expressed in the same units.

us_r,

C O N C E P T U A L O B J E C T I V E S

■

Understand that degrees and radians are both

measures of angles.

■

Realize that radian measure allows us to write

trigonometric functions as functions of real numbers.

R A D I A N M E A S U R E

S E C T I O N

3.1

S K I L L S O B J E C T I V E S

■

Calculate the radian measure of an angle.

■

Convert between degrees and radians.

■

Calculate trigonometric function values for angles

given in radians.

s1 s2 ␪1 ␪2 r r r r

If

radian, then the length of the subtended arc is equal to the radius,

This

leads to a general definition of

radian measure

.

s

1

r

.

u

1

1

If a central angle in a circle with radius

r

intercepts

an arc on the circle of length

s

, then the measure of

in

radians, is given by

Note:

The formula is valid only if

s

(arc length) and

r

(radius) are expressed in the same units.

u

(in radians)

s

r

u

,

u

Radian Measure

D

E F I N I T I O N s ␪ r r 130

3.1 Radian Measure 131

One full rotation corresponds to an arc length equal to the circumference

of the

circle with radius

r

. We see then that one full rotation is equal to

radians.

u

full rotation

2

p

r

r

2

p

2

p

2

p

r

■Answer:0.3 rad E X A M P L E 1

Finding the Radian Measure of an Angle

What is the measure (in radians) of a central angle that intercepts an arc of length 4 feet on a circle with radius 10 feet?

Solution:

Write the formula relating radian measure to arc length and radius.

Let and

■Y O U R T U R N What is the measure (in radians) of a central angle ␪that intercepts an arc of length 3 inches on a circle with radius 50 inches?

u 4 ft 10 ft 0.4 rad r10 feet. s4 feet us_r u ■Answer:0.06 rad

E X A M P L E 2

Finding the Radian Measure of an Angle

What is the measure (in radians) of a central angle that intercepts an arc of length 6 centimeters on a circle with radius 2 meters?

u

C

O M M O N

M

I S T A K E

A common mistake is forgetting to first put the radius and arc length in the same units.

C O R R E C T

Write the formula relating radian measure to arc length and radius.

Substitute and into the radian expression.

Convert the radius (2) meters to centimeters:

The units, centimeters, cancel and the result is a unitless real number.

u0.03 rad u 6 cm 200 cm 2 meters200 centimeters u6 cm 2 m r2 meters s6 centimeters u (in radians)s r I N C O R R E C T Substitute and into the radian expression.

ERROR(not converting both numerator and denominator to the same units)

3 u6 cm_{2 m} r2 meters

s6 centimeters

★

■Y O U R T U R N What is the measure (in radians) of a central angle that intercepts

an arc of length 12 millimeters on a circle with radius 4 centimeters?

u

C A U T I O N

Units for arc length and radius must be the same in order to use

us r Study Tip

Notice in Example 1 that the units, feet, cancel, therefore leaving as a unitless real number, 0.4.

u

Classroom Example 3.1.1 Find the measure, in radians, of the central angle that intercepts an arc of length 3 yards on a circle of radius 6 yards.

Answer:12rad

u

Classroom Example 3.1.2 Find the measure, in radians, of the central angle that intercepts an arc of length 3 yards on a circle of radius 6 feet.

Answer:32rad

u

Because radians are unitless, the word radians (or rad) is often omitted. If an angle

measure is given simply as a real number, then radians are implied.

WOR DS MATH

The measure of is 4 degrees.

The measure of is 4 radians.

Converting Between Degrees and Radians

To convert between degrees and radians, we must first look for a relationship between

them. We start by considering one full rotation around the circle. An angle corresponding

to one full rotation is said to have measure

, and we saw previously that one full

rotation corresponds to

rad.

WOR DS MATH

Write the angle measure (in degrees) that

corresponds to one full rotation.

Write the angle measure (in radians) that

corresponds to one full rotation.

Arc length is the circumference of the circle.

Substitute into

Equate the measures corresponding to one

full rotation.

Divide by 2.

Divide by 180° or

␲

.

1

p

180°

or

1

180°

p

180°

p

rad

360°

2

p

rad

u

2

p

r

r

2

p

rad

u

(in radians)

s

r

.

s

2

p

r

s

2

p

r

u

360°

u

2

p

360°

u

4

u

u

4°

u

This leads us to formulas

that convert between degrees

and radians. Let

represent an angle measure given in degrees and

represent the

corresponding angle measure given in radians.

u

r

u

d

a

unit conversations, like

1 hr

60 min

b

To convert degrees to radians, multiply the degree measure by

u

r

u

d

a

p

180°

b

p

180°

.

C

O N V E R T I N G D E G R E E S TO R A D I A N S

To convert radians to degrees, multiply the radian measure by

u

d

u

r

a

180°

p

b

180°

p

.

C

O N V E R T I N G R A D I A N S TO D E G R E E S

Before we begin converting between degrees and radians, let’s first get a feel for

radians. How many degrees is 1 radian?

WOR DS MATH

Multiply 1 radian by

Approximate by

3.14.

Use a calculator to evaluate and

round to the nearest degree.

A radian is much larger than a degree (almost 57 times larger). Let’s compare two

angles, one measuring 30 radians and the other measuring

Note that

revolutions, whereas revolution.

x y 30º x y 30 rad

30°

121

30 rad

2

p

rad/rev

⬇

4.77

30°.

1 rad

⬇

57°

⬇

57°

1

a

180°

3.14

b

p

1

a

180°

p

b

180°

p

.

3.1 Radian Measure 133 ■Answer:p or 1.047 3

E X A M P L E 3

Converting Degrees to Radians

Convert to radians.

Solution:

Multiply by Simplify.

Note: is the exact value. A calculator can be used to approximate this expression. Scientific and graphing calculators have a button. The decimal approximation of rounded to three decimal places is 0.785.

Exact Value: Approximate Value: ■Y O U R T U R N Convert to 60° radians. 0.785 p 4 p 4 p p 4 p 4 rad (45°)a p 180°b 45°p 180° p 180°. 45° 45° Classroom Example 3.1.3 Convert to radians. Answer:3p 4 135° c03.qxd 8/22/11 7:07 PM Page 133

E X A M P L E 4

Converting Degrees to Radians

Convert to radians.

Solution:

Multiply by

Simplify (factor out the common 4).

Use a calculator to approximate.

■Y O U R T U R N Convert to 460° radians. ⬇ 8.238 rad 118₄₅ p 472°a p 180°b p 180°. 472° 472° ■Answer:270° ■Answer:23 or 8.029 9p

E X A M P L E 5

Converting Radians to Degrees

Convert to degrees. Solution: Multiply by Simplify. ■ Y O U R T U R N Convert to 3p degrees. 2 120° 2p 3 ⴢ 180° p 180° p . 2p 3 2p 3

E X A M P L E 6

Converting Radians to Degrees

Convert 10 radians to degrees.

Solution: Multiply 10 radians by . Simplify. 1800° p ⬇ 573° 10ⴢ180° p 180° p Classroom Example 3.1.5 Convert to degrees. Answer:330° 11p 6 Classroom Example 3.1.4 a.* Convert to radians, where nis an integer. b. Convert to radians. Answer: a. b. 100p 9 (2n1)p 2000° 180(2n1)°

Since

, we know the following special angles:

and we can now draw the unit circle with the special angles in both

degrees

and

radians

.

p

6

30°

p

4

45°

p

3

60°

p

2

90°

p

180°

60º = 3 ␲ 45º = 4 ␲ 30º = 6 ␲ 360º = 2␲ 330º = 6 11␲ 315º = 4 7␲ 300º = 3 5␲ 270º = 2 3␲ 240º = 3 4␲ 225º = 4 5␲ 210º = 6 7␲ 180º = ␲ 150º = 6 5␲ 135º = 4 3␲ 90º = 2 ␲ 120º = 3 2␲

E X A M P L E 7

Evaluating Trigonometric Functions

for Angles in Radian Measure

Evaluate exactly.

Solution:

Recognize that or convert to degrees. Find the value of

Equate and sin

■Y O U R T U R N Evaluate exactly.cosap 3b sinap 3b 13 2 ap₃b. sin60° sin60° 13 2 sin60°. p 3 ⴢ 180° p 60° p 3 p 360° sinap 3b

The following table lists sine and cosine values for special angles in both degrees and

radians. Tangent, secant, cosecant, and cotangent values can all be found from sine

and cosine values using quotient and reciprocal identities. The table only lists special

angles in quadrant I and quadrantal angles (

or

). Values in

quadrants II, III, and IV can be found using reference angles and knowledge of the

algebraic sign

(

or

)

of the sine and cosine functions in each quadrant.

0

u

2

p

0°

u

360°

3.1 Radian Measure 135 ■Answer:1 2 Technology Tip Set a TI/scientific calculator to radian mode by typing

. (radian) ENTER

䉲 䉲

MODE

Use a TI/scientific calculator to check the value of and . Press .2nd _^ p

a13 2 b sinap

3b

If the angle of the trigonometric function to be evaluated has its terminal side in

quadrants II, III, or IV, then we use reference angles and knowledge of the algebraic sign

in that quadrant. We know how to find reference angles in degrees. Now we will

find reference angles in radians.

(

or

)

VALUE OF

ANGLE, ␪ TRIGONOMETRIC FUNCTION

RADIANS DEGREES SIN␪ COS␪

0 0° 0 1 30° 45° 60° 90° 1 0 180° 0 270° 0 360° 0 1 2p 1 3p 2 1 p p 2 1 2 13 2 p 3 12 2 12 2 p 4 13 2 1 2 p 6 c03.qxd 8/22/11 7:07 PM Page 135

■Answer:p 3

E X A M P L E 8

Finding Reference Angles in Radians

Find the reference angle for each angle given.

a. b.

Solution (a):

The terminal side of lies in quadrant II. Recall that radians is of a full revolution, so is of a half of revolution.

The reference angle is made with the terminal side and the negative x-axis. Solution (b):

The terminal side of lies in quadrant IV. Recall that is a complete revolution.

Note that is not quite .

The reference angle is made with the terminal side and the positive x-axis.

■Y O U R T U R N Find the reference angle for 5p 3. 2p11p 6 12p 6 11p 6 p 6 aor 12p 6 b 2p 11p 6 2p u p3p 4 4p 4 3p 4 p 4 3 4 3 4p 1 2 p u 11p 6 3p 4 Classroom Example 3.1.8

Find the reference angle for each angle given.

a. b. Answer: a. b. p 4 p 3 5p 4 2p 3 x y 3␲ 4 ␣ x y 11␲ 6 ␣

TERMINAL SIDELIES IN . . . DEGREES RADIANS QI QII QIII QIV a360°u a2pu aup au180° apu a180°u au au

E X A M P L E 9

Evaluating Trigonometric Functions for Angles

in Radian Measure Using Reference Angles

Evaluate exactly.

Solution:

The terminal side of angle lies in

quadrant III since .

The reference angle is

Find the cosine value for the reference angle. Determine the algebraic sign for the cosine

function in quadrant III. Negative

Combine the algebraic sign of the cosine function in quadrant III with the value of the cosine function of the reference angle. Confirm with a calculator.

■Y O U R T U R N Evaluate exactly.sina7p 4 b 0.707⬇0.707 cosa5p 4b 12 2 () cosap 4bcos45° 12 2 p 4 45°. 5p 4 p p 4 5p 4 cosa5p 4 b Technology Tip Use the TI/scientific calculator to check the value for and compare with 12 2 . cosa5p 4b ■Answer:12 2 x y = 45º 4 ␲ 4 5␲

One radian is approximately equal to Careful attention must be paid to what mode (degrees or radians) calculators are set when evaluating trigonometric functions. To evaluate a trigonometric function for nonacute angles in radians, we use reference angles (in radians) and knowledge of the algebraic sign of the trigonometric function.

57°.

S U M M A R Y

In this section, a second measure of angles was introduced, which allows us to write trigonometric functions as functions of real numbers. A central angle of a circle has radian measure equal to the ratio of the arc length intercepted by the angle to the radius of the circle, .

Radians and degrees are related by the relation that ■ To convert from radians to degrees, multiply the

radian measure by

■ To convert from degrees to radians, multiply the degree measure by p 180°. 180° p . p180°. us_r S E C T I O N

3.1

3.1 Radian Measure 137 Classroom Example 3.1.9 Evaluate exactly. Answer:13 2 cosa5p 6 b c03.qxd 8/22/11 7:07 PM Page 137

■_{S K I L L S}

E X E R C I S E S

S E C T I O N

3.1

In Exercises 1–10, find the measure (in radians) of a central angle that intercepts an arc on a circle of radius rwith indicated arc length s.

␪ 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

In Exercises 11–24, convert each angle measure from degrees to radians. Leave answers in terms of

11. 12. 13. 14. 15. 16. 17.

18. 19. 20. 21. 22. 23. 24.

In Exercises 25–38, convert each angle measure from radians to degrees.

25. 26. 27. 28. 29. 30. 31.

32. 33. 34. 35. 36. 37. 38.

In Exercises 39–44, convert each angle measure from radians to degrees. Round answers to the nearest hundredth of a degree.

39. 4 40. 3 41. 0.85 42. 3.27 43. 44.

In Exercises 45–50, convert each angle measure from degrees to radians. Round answers to three significant digits.

45. 46. 47. 48. 49. 50.

In Exercises 51–58, find the reference angle for each of the following angles in terms of both radians and degrees.

51. 52. 53. 54. 55. 56. 57. 58.

In Exercises 59–84, find the exactvalue of the following expressions. Do not use a calculator.

59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. sina8p 3 b cosa17p 6 b cosa11p 3 b sina13p 4 b cota3p 2 b sec(5p) cscap 2b cota3p 2 b tana3p 4 b tana5p 6 b tana5p 6 b tanap 6b tana5p 3 b tana11p 6 b sina5p 6 b cosa5p 3 b cosap 4b sinap 6b cosa11p 6 b sina4p 3 b cosa7p 6b sina3p 4 b cosa2p 3 b sina7p 4 b cosap 6b sinap 4b 9p 4 4p 3 7p 12 5p 12 5p 4 7p 4 3p 4 2p 3 298.7° 56.5° 172° 112° 65° 47° 5.9841 2.7989 8p 9 7p 15 13p 36 19p 20 6p 9p 7p 3 5p 12 11p 9 3p 8 7p 6 3p 4 p 4 p 6 320° 210° 540° 780° 340° 170° 100° 75° 270° 315° 90° 45° 60° 30° ␲. s0.2 mm r1.6 cm, s5 mm r2.5 cm, s 3 14 cm r3 4 cm, s 1 32 in. r1 4 in., s2 cm r1 m, s20 mm r100 cm, s1 in. r6 in., s4 in. r22 in., s2 cm r20 cm, s2 cm r10 cm,

Kin Cheu

ng/R

euters/Lando

v

For Exercises 85 and 86, refer to the following:

Two electronic signals that are not co-phased are called out of phase. Two signals that cancel each other out are said to be

out of phase, or the difference in their phases is 85. Electronic Signals.How many radians out of phase are

two signals whose phase difference is

86. Electronic Signals.How many radians out of phase are two signals whose phase difference is

87. Construction.In China, you find circular clan homes called tulou. Some tulou are three or four stories high and exceed 70 meters in diameter. If a wedge or section on the third floor of such a building has a central angle measuring

how many radians is this? 36°,

110°? 270°?

180°. 180°

93. Sprinkler. A water sprinkler can reach an arc of 15 feet, 20 feet from the sprinkler as shown. Through how many radians does the sprinkler rotate?

94. Sprinkler. A sprinkler is set to reach an arc of 35 feet, 15 feet from the sprinkler. Through how many radians does the sprinkler rotate?

95. Engine. If a car engine is said to be running at 1500 RPMs (revolutions per minute), through how many radians is the engine turning every second?

96. Engine. If a car engine is said to rotate per second, through how many radians does the engine turn each second? For Exercises 97 and 98, refer to the following:

A traction splint is commonly used to treat complete long bone fractures of the leg. The angle between the leg and torso is an oblique angle . The reference angle is the acute angle between the leg in traction and the bed.

97. Health/Medicine. If find the measure of the reference angle in both radians and degrees. 98. Health/Medicine. If find the measure of the

reference angle in both radians and degrees. u⫽2₃p, u⫽3p 4, ␣ ␪ a u 15,000° 15 ft 20 ft ■A P P L I C A T I O N S

88. Construction.In China, you find circular clan homes called tulou. Some tulou are three or four stories high and exceed 70 meters in diameter. If a wedge or section on the third floor of such a building has a central angle measuring how many radians is this?

89. Clock.How many radians does the second hand of a clock turn in minutes?

90. Clock.How many radians does the second hand of a clock turn in 3 minutes and 15 seconds?

91. London Eye.The London Eye has 32 capsules (each capable of holding 25 passengers with an unobstructed view of London). What is the radian measure of the angle made between the center of the wheel and the spokes aligning with each capsule?

92. Space Needle.The space needle in Seattle has a restaurant that offers views of Mount Rainier and Puget Sound. The restaurant completes one full rotation in approximately 45 minutes. How many radians will the restaurant have rotated in 25 minutes?

212

72°,

3.1 Radian Measure 139

In Exercises 103–106, explain the mistake that is made. 105. Evaluate . Solution: Evaluate and Substitute the values of the trigonometric functions. Simplify.

This is incorrect. What mistake was made? 106. Approximate with a calculator

Round to three decimal places. Solution:

Evaluate the trigonometric functions individually. Substitute the values into the expression. Simplify.

This is incorrect. What mistake was made? cos(42)tan(65)sin(12)⬇2.680

cos(42)tan(65)sin(12)⬇0.7432.1450.208 sin(12)⬇0.208 tan(65)⬇2.145

cos(42)⬇0.743

cos(42)tan(65)sin(12). 6tan(45)5secap 3b16 6tan(45)5secap 3b6(1)5(2) secap 3b2 tan(45)1 secap 3b. tan(45) 6tan(45)5secap 3b ■_{C A T C H T H E M I S T A K E}

103. What is the measure (in radians) of a central angle that intercepts an arc of length 6 centimeters on a circle with radius 2 meters?

Solution:

Write the formula for radians. Substitute

Write the angle in terms of radians. rad This is incorrect. What mistake was made?

104. What is the measure (in radians) of a central angle that intercepts an arc of length 2 inches on a circle with radius 1 foot?

Solution:

Write the formula for radians. Substitute

Write the angle in terms of radians. This is incorrect. What mistake was made?

u2 rad u2 1 s2, r1. us_r u u3 u6 2 s6, r2. u For Exercises 99–102, refer to the following:

A water molecule is composed of one oxygen atom and two hydrogen atoms and exhibits a bent shape with the oxygen atom at the center.

99. Chemistry.The angle between the O-H bonds in a water molecule is approximately 105. Find the angle between the O-H bonds of a water molecule in radians.

105º +

– Attraction of bonding electrons to the oxygen creates local negative and positive particle charges

Net positive charge Net negative charge

␦ ␦ + ␦ O O O O O O O O O O H H

100. Chemistry.The angle between the S-O bonds in sulfur dioxide (SO2) is approximately 120. Find the angle between the S-O bonds of sulfur dioxide in radians. 101. Chemistry/Environment.Nitrogen dioxide (NO2) is a

toxic gas and prominent air pollutant. The angle between the N-O bond is 134.3. Find the angle between the N-O bonds in radians.

102. Chemistry/Environment.Methane (CH4) is a chemical compound and potent greenhouse gas. The angle between the C-H bonds is 109.5°. Find the angle between the C-H bonds in radians. C H H H H 108.70 pm 109.5 º 134.3º O O 119.7 pm N

■T E C H N O L O G Y

120. With a calculator set in radian mode, find With a calculator set in degree mode, find Why do your results make sense?

cosa5180° p b. cos5. 119. With a calculator set in radian mode, find With a

calculator set in degree mode, find Why do your results make sense?

sina42180° p b. sin42.

C O N C E P T U A L O B J E C T I V E

■

Understand that to use the arc length formula, the

angle measure must be in radians.

A R C L E N G T H A N D A R E A O F

A C I R C U L A R S E C T O R

S E C T I O N

3.2

S K I L L S O B J E C T I V E S

■

Calculate the length of an arc along a circle.

■

Find the area of a circular sector.

■

Solve application problems involving circular arc

lengths and sectors.

In Section 3.1, radian measure was defined in terms of the ratio of a circular arc of length

s

and length of the circle’s radius

r

.

In this section (3.2) and the next (3.3), we look at applications of radian measure that

involve calculating

arc lengths

and

areas of circular sectors

and calculating

angular and

linear speeds

.

u

(in radians)

s

r

3.2 Arc Length and Area of a Circular Sector 141

■C O N C E P T U A L

In Exercises 107–110, determine whether each statement is true or false.

113. The distance between Atlanta, Georgia, and Boston, Massachusetts, is approximately 900 miles along the curved surface of the Earth. The radius of the Earth is approximately 4000 miles. What is the central angle with vertex at the center of the Earth and sides of the angles intersecting the surface of the Earth in Atlanta and Boston? 114. The radius of the Earth is approximately 6400 kilometers.

If a central angle, with vertex at the center of the Earth, intersects the surface of the Earth in London (UK) and Rome (Italy) with a central angle of 0.22 radians, what is the distance along the Earth’s surface between London and Rome? Round to the nearest hundred kilometers. ■C H A L L E N G E

115. At 8:20, what is the radian measure of the smaller angle between the hour hand and minute hand?

116. At 9:05, what is the radian measure of the larger angle between the hour hand and minute hand?

117. Find the exact value for

for . 118. Find the exact value for

for x p. 2cosa3xp 3b2sina x 6b5 xp 3 5cosa3xp 2b2sin(2x) 5

110. The sum of the angles with radian measure in a triangle is

111. Find the sum of complementary angles in radian measure. 112. How many complete revolutions does an angle with

measure 92 radians make? p.

107. An angle with measure 4 radians is a quadrant II angle. 108. Angles expressed exactly in radian measure are always

given in terms of .

109. For an angle with positive measure, it is possible for the numerical values of the degree and radian measures to be equal.

p c03.qxd 8/22/11 7:08 PM Page 141

Study Tip To use the relationship

the angle must be in radians.u

sru

Arc Length

From geometry we know the length of an arc of a circle is proportional to its central angle.

In Section 3.1, we learned that for the special case when the arc length is equal to the

circumference of the circle, the angle measure in radians corresponding to one full rotation is

Let us now assume that we are given the central angle and we want to find the arc length.

WOR DS MATH

Write the definition of radian measure.

Multiply both sides of the equation by

r

.

Simplify.

The formula

is true only when is in radians. To develop a formula when is in

degrees, we multiply by

p

to convert the angle measure to radians.

180°

u

u

u

s

r

u

r

u

s

r

ⴢ

u

s

r

ⴢ

r

u

s

r

2

p

.

E X A M P L E 1

Finding Arc Length When the Angle

Has Radian Measure

In a circle with radius 10 centimeters, an arc is intercepted by a central angle with measure Find the arc length.

Solution:

Write the formula for arc length when the angle has radian measure.

Substitute and

Simplify.

■Y O U R T U R N In a circle with radius 15 inches, an arc is intercepted by a central angle with measure . Find the arc length.p

3 s 35p 2 cm s(10 cm)a7p 4 b ur 7p 4 . r10 centimeters srur 7p 4.

If a central angle in a circle with radius

r

intercepts an arc on the circle of length

s

, then the arc length

s

is given by

u

Arc Length

D

E F I N I T I O N

■Answer:5pin.

Classroom Example 3.2.1 Find the arc length of a sector determined by central angle

on a circle with radius 24 meters. Answer:44p m 11p 6

is in radians.

is in degrees.

u

d

s

r

u

d

a

p

180°

b

u

r

s

r

u

r

3.2 Arc Length and Area of a Circular Sector 143

E X A M P L E 2

Finding Arc Length When the Angle

Has Degree Measure

In a circle with radius 7.5 centimeters, an arc is intercepted by a central angle with measure Find the arc length. Approximate the arc length to the nearest centimeter. Solution:

Write the formula for arc length when the angle has degree measure.

Substitute and Evaluate the result with a calculator.

Round to the nearest centimeter.

■ Y O U R T U R N In a circle with radius 20 meters, an arc is intercepted by a central angle with measure Find the arc length. Approximate the arc length to the nearest meter.

113°. s⬇10 cm s⬇9.948 cm s(7.5 cm) (76°)a p 180°b ud76°. r7.5 centimeters sruda p 180°b

76°.

E X A M P L E 3

Path of International Space Station

The International Space Station (ISS) is in an

approximate circular orbit 400 kilometers above the surface of the Earth. If the ground station tracks the space station when it is within a central angle of this circular orbit about the center of the Earth above the tracking antenna, how many kilometers does the ISS cover while it is being tracked by the ground station? Assume that the radius of the Earth is 6400 kilometers. Round to the nearest kilometer.

Solution:

Write the formula for arc length when the angle has degree measure.

Recognize that the radius of the orbit is and that

Evaluate with a calculator. Round to the nearest kilometer.

The ISS travels approximately 5341 kilometers during the ground station tracking.

■Y O U R T U R N If the ground station in Example 3 could track the ISS within a

central angle of its circular orbit about the center of the Earth, how far would the ISS travel during the ground station tracking?

60° s⬇5341 km s⬇5340.708 km ud45°. s(6800 km) (45°)a p 180°b 4006800 kilometers r6400 srud a p 180°b 45° ■Answer:39 m ■Answer:7121 km 400 km 6400 km 45º ISS c03.qxd 8/22/11 7:08 PM Page 143

E X A M P L E 4

Gears

Gears are inside many devices like automobiles and power meters. When the smaller gear drives the larger gear, then typically the driving gear is rotated faster than a larger gear would be if it were the drive gear. In general, smaller ratios of radius of the driving gear to the driven gear are called for when machines are expected to yield more power. The smaller gear has a radius of 3 centimeters, and the larger gear has a radius of 6.4 centimeters. If the smaller gear rotates how many degrees has the larger gear rotated? Round the answer to the nearest degree.

Solution:

Recognize that the small gear arc lengththe large gear arc length. Smaller Gear

Write the formula for arc length when the angle has degree measure.

Substitute the values for the smaller gear: and

Simplify. Larger Gear

Remember that the larger gear’s arc length is equal to the smaller gear’s arc length. Write the formula for arc length when the angle has degree measure.

Substitute and

Solve for Simplify.

Round to the nearest degree.

The larger gear rotates approximately 80°.

ud80° ud⬇79.6875° ud 180° p ⴢ 17p cm 6(6.4 cm) ud. a17₆p cmb(6.4 cm)uda p 180°b sa17p 6 b centimeter srud a p 180°b sa17p 6 b cm ssmallera 17p 6 b cm ud170°. r3 centimeters ssmaller(3 cm) (170°)a p 180°b srud a p 180°b 6.4 cm 3 cm 170°,

Area of a Circular Sector

A restaurant lists a piece of French silk pie as having 400 calories. How does the chef

arrive at that number? She calculates the calories of all the ingredients that went into

making the entire pie and then divides by the number of slices the pie yields. For

example, if an entire pie has 3200 calories and it is sliced into 8 equal pieces, then each

Technology Tip

When solving for be sure to use a pair of parentheses for the product in the denominator.

180°ⴢ17 6(6.4) ud 180° p ⴢ 17p cm 6(6.4 cm) ud, Study Tip

Notice that when calculating in Example 4, the centimeter units cancel but its degree measure remains.

ud Classroom Example 3.2.4 Consider two gears working together such that the smaller gear has a radius of 10 centimeters, while the larger gear has a radius measuring 25 centimeters. Through how many degrees does the small gear rotate when the large gear makes one complete rotation? Answer:900°

piece has 400 calories. Although that example involves volume, the idea is the same

with areas of sectors of circles.

Circular sectors

can be thought of as “pieces of a pie.”

Recall that arc lengths of a circle are proportional to the central angle (in radians) and

the radius. Similarly, a circular sector is a portion of the entire circle. Let

A

represent the

area of the

sector of the circle

and

represent the central angle (in radians) that forms

the sector. Then, let us consider the entire circle whose area is

and the angle that

represents one full rotation has measure

(radians).

WOR DS MATH

Write the ratio of the area of the sector to the

area of the entire circle.

Write the ratio of the central angle

␪

r

to the

measure of one full rotation.

The ratios must be equal (proportionality of

sector to circle).

Multiply both sides of the equation by

Simplify.

A

1

2

r

2

u

r

p

r

2

ⴢ

A

p

r

2

u

r

2

p

ⴢ

p

r

2

p

r

2

.

A

p

r

2

u

r

2

p

u

r

2

p

A

p

r

2

2

p

p

r

2

u

r

3.2 Arc Length and Area of a Circular Sector 145

s ␪ r r 〈 Study Tip To use the relationship

the angle ␪must be in radians. A1

2r2u

The area of a sector of a circle

with radius

r

and central angle is given by

u

Area of a Circular Sector

D

E F I N I T I O N

E X A M P L E 5

Finding the Area of a Circular Sector When

the Angle Has Radian Measure

Find the area of the sector associated with a single slice of pizza if the entire pizza has a 14-inch diameter and the pizza is cut into 8 equal pieces.

Solution:

The radius is half the diameter.

Find the angle of each slice if the pizza is cut into 8 pieces ( of the complete revolution). Write the formula for circular sector area

in radians. A 1 2r 2_u r ur 2p 8 p 4 2p 1 8 r14 2 7 in. Classroom Example 3.2.5 Find the area of the sector with diameter 16 feet and

central angle . Answer:28p ft2 7p 8

is in radians.

is in degrees.

u

d

A

1

2

r

2

u

d

a

p

180°

b

u

r

A

1

2

r

2

u

r c03.qxd 8/22/11 7:08 PM Page 145

■Answer:8p in.2_⬇25in.2

Substitute and into

the area equation. Simplify.

Approximate the area with a calculator.

■Y O U R T U R N Find the area of a slice of pizza (cut into 8 equal pieces) if the entire pizza has a 16-inch diameter.

A⬇19 in.2 A49p 8 in. 2 A1 2(7 in.) 2_ap 4b ur p 4 r7 inches

E X A M P L E 6

Finding the Area of a Circular Sector When

the Angle Has Degree Measure

Sprinkler heads come in all different sizes depending on the angle of rotation desired. If a sprinkler head rotates and has enough pressure to keep a constant 25-foot spray, what is the area of the sector of the lawn that gets watered? Round to the nearest square foot. Solution:

Write the formula for circular sector area in degrees.

Substitute r25 feet and ␪d90 into the area equation.

Simplify.

Round to the nearest square foot.

■Y O U R T U R N If a sprinkler head rotates and has enough pressure to keep a constant 30-foot spray, what is the area of the sector of the lawn it can water? Round to the nearest square foot.

180° A⬇491 ft2 Aa625p 4 b ft 2_{⬇490.87 ft}2 A1 2(25 ft) 2_(90°)a p 180°b A1 2r 2_u d a p 180°b 90° ■Answer:450p ft2_⬇1414ft2 SMH

The formula for the area of a sector of a circle was also developed for the cases in which the central angle is given in either radians or degrees.

S U M M A R Y

In this section, we used the proportionality concept (both the arc length and area of a sector are proportional to the central angle of a circle). The definition of radian measure was used to develop formulas for the arc length of a circle when the central angle is given in either radians or degrees.

S E C T I O N

3.2

Classroom Example 3.2.6 Find the exact area of the sector with diameter 1.4 inches

and central angle .

Answer:49p 160 in. 2 225° is in radians. is in degrees. ud srud a p 180°b ur srur is in radians. is in degrees. ud A1 2r 2_u d a p 180°b ur A1 2r 2_u r

In Exercises 1–12, find the exact length of each arc made by the indicated central angle and radius of each circle.

■S K I L L S

E X E R C I S E S

S E C T I O N

3.2

In Exercises 13–24, find the exact length of each radius given the arc length and central angle of each circle.

In Exercises 25–36, use a calculator to approximate the length of each arc made by the indicated central angle and radius of each circle. Round answers to two significant digits.

In Exercises 37–48, find the area of the circular sector given the indicated radius and central angle. Round answers to three significant digits. 1. 2. 3. 4. 5. , 6. in. 7. 8. 9. u8°,r1500 km 10. u3°,r1800 km 11. u48°, r24cm 12. u30°, r120cm r15 ␮m u14°, r18 ␮m u22°, up 4, r10 r3.5 m u2p 7 r6 yd up 8, r8 ft u p 12, r5 cm u4, r4 mm u3, 13. 14. 15. 16. 17. yd, 18. in., 19. 20. 21. 22. 23. km, 24. s3p ft,u35o 16 u45o s2p 11 u30° sp 4␮m, u40° s8p 3 mi, u15° s11p 6 cm, u20° s4p 9 yd, u3p 2 s 4p u4p 5 s 12p 5 u₁₈₀p s5p 9 km, u3₅p s24p 5 in., u₁₂p s5p 6 m, u₁₀p s5p 2 ft, 37. 38. 39. 40. 41. cm 42. m 43. 44. 45. u1.2°,r1.5 ft 46. u14°,r3.0 ft 47. u22.8o, r2.6mi 48. u60°, r15km r2.5 mm u27°, r4.2 cm u56°, u2₃p, r33 u3₁₁p, r10 r13 mi u5p 6, r2.2 km u3p 8, r3 in. up 5, r7 ft up 6, 25. 26. 27. 28. 29. mi 30. mm 31. 32. 33. u29°, r2500 km 34. u11°, r2200 km 35. u57°, r22ft 36. u127°, r58in. r0.63 ␮m u19.7°, r1.55 ␮m u79.5°, u7₈p, r17 r30 u4.95, r6 ft u₁₀p, r8 yd u₁₅p, r5.5 cm u2.4, r0.4 mm u3.3,

3.2 Arc Length and Area of a Circular Sector 147

Dav id B a ll/Index Stoc k/Photolibr ar y

49. Low Earth Orbit Satellites.A low Earth orbit (LEO) satellite is in an approximate circular orbit 300 kilometers above the surface of the Earth. If the ground station tracks the satellite when it is within a cone above the tracking antenna (directly overhead), how many kilometers does the satellite cover during the ground station track? Assume the radius of the Earth is 6400 kilometers. Round your answer to the nearest kilometer.

50. Low Earth Orbit Satellites.A low Earth orbit (LEO) satellite is in an approximate circular orbit 250 kilometers above the surface of the Earth. If the ground station tracks the satellite when it is within a cone above the tracking antenna (directly overhead), how many kilometers does the satellite cover during the ground station track? Assume the radius of the Earth is 6400 kilometers. Round your answer to the nearest kilometer.

51. Big Ben.The famous clock tower in London has a minute hand that is 14 feet long. How far does the tip of the minute hand of Big Ben travel in 25 minutes? Round your answer to the nearest foot.

52. Big Ben.The famous clock tower in London has a minute hand that is 14 feet long. How far does the tip of the minute hand of Big Ben travel in 35 minutes? Round your answer to two decimal places.

53. London Eye.The London Eye is a wheel that has 32 capsules and a diameter of 400 feet. What is the distance someone has traveled once they reach the highest point for the first time?

30° 45°

54. London Eye.Assuming the wheel stops at each capsule in Exercise 53, what is the distance someone has traveled from the point he or she first gets in the capsule to the point at which the Eye stops for the sixth time during the ride? 55. Gears.The smaller gear shown below has a radius of

5 centimeters, and the larger gear has a radius of 12.1 centimeters. If the smaller gear rotates how many degrees has the larger gear rotated? Round the answer to the nearest degree.

56. Gears.The smaller gear has a radius of 3 inches, and the larger gear has a radius of 15 inches (see the figure above). If the smaller gear rotates how many degrees has the larger gear rotated? Round the answer to the nearest degree.

57. Bicycle Low Gear. If a bicycle has 26-inch diameter wheels, the front chain drive has a radius of 2.2 inches, and the back drive has a radius of 3 inches, how far does the bicycle travel for every one rotation of the cranks (pedals)? 420°, 120°, Getty Im ag es, Inc. ■A P P L I C A T I O N S

58. Bicycle High Gear.If a bicycle has 26-inch diameter wheels, the front chain drive has a radius of 4 inches, and the back drive has a radius of 1 inch, how far does the bicycle travel for every one rotation of the cranks (pedals)?

59. Odometer.A Ford Expedition Eddie Bauer Edition comes standard with -inch rims (which corresponds to a tire with -inch diameter). Suppose you decide to later upgrade these tires for -inch rims (corresponding to a tire with -inch diameter). If you do not get your onboard computer reset for the new tires, the odometer will not be accurate. After your new tires have actually driven 1000 miles, how many miles will the odometer report the Expedition has been driven? Round to the nearest mile. 60. Odometer.For the same Ford Expedition Eddie Bauer

Edition in Exercise 59, after you have driven 50,000 miles, how many miles will the odometer report the Expedition has been driven if the computer is not reset to account for the new oversized tires? Round to the nearest mile.

61. Sprinkler Coverage.A sprinkler has a 20-foot spray and covers an angle of What is the area that the sprinkler waters?

62. Sprinkler Coverage.A sprinkler has a 22-foot spray and covers an angle of What is the area that the sprinkler waters?

63. Windshield Wiper.A windshield wiper that is 12 inches long (blade and arm) rotates If the rubber part is 8 inches long, what is the area cleared by the wiper? Round to the nearest square inch.

64. Windshield Wiper.A windshield wiper that is 11 inches long (blade and arm) rotates If the rubber part is 7 inches long, what is the area cleared by the wiper? Round to the nearest square inch.

65. Bicycle Wheel.A bicycle wheel 26 inches in diameter travels in 0.05 seconds. Through how many revolutions does the wheel turn in 30 seconds?

66. Bicycle Wheel.A bicycle wheel 26 inches in diameter travels in 0.075 seconds. Through how many revolutions does the wheel turn in 30 seconds?

2p 3 45° 65°. 70°. 60°. 45°. 28.2 19 25.7 17

67. Bicycle Wheel.A bicycle wheel 26 inches in diameter travels 20 inches in 0.10 seconds. What is the speed of the wheel in revolutions per second?

68. Bicycle Wheel.A bicycle wheel 26 inches in diameter travels at four revolutions per second. Through how many radians does the wheel turn in 0.5 seconds?

For Exercises 69 and 70, refer to the following:

Sniffers outside a chemical munitions disposal site monitor the atmosphere surrounding the site to detect any toxic gases. In the event that there is an accidental release of toxic fumes, the data provided by the sniffers make it possible to determine both the distance dthat the fumes reach as well as the angle of spread that sweep out a circular sector.

69. Environment.If the maximum angle of spread is 105° and the maximum distance at which the toxic fumes were detected was 9 miles from the site, find the area of the circular sector affected by the accidental release. 70. Environment.To protect the public from the fumes,

officials must secure the perimeter of this area. Find the perimeter of the circular sector in Exercise 69.

For Exercises 71 and 72, refer to the following:

The structure of human DNA is a lineardouble helix formed of nucleotide base pairs (two nucleotides) that are stacked with spacing of 3.4 angstroms (3.41012m), and each base pair is rotated 36with respect to an adjacent pair and has 10 base pairs per helical turn. The DNA of a virus or a bacterium, however, is a circulardouble helix (see the figure below) with the structure varying among species.

(Source: http://www.biophysics.org/Portals/1/ PDFs/Education/Vologodskii.pdf.)

71. Biology.If the circular DNA of a virus has 10 twists (or turns) per circle and an inner diameter of 4.5 nanometers, find the arc length between consecutive twists of the DNA. 72. Biology.If the circular DNA of a virus has 40 twists (or

turns) per circle and an inner diameter of 2.0 nanometers, find the arc length between consecutive twists of the DNA.

Twists

u 3.2 Arc Length and Area of a Circular Sector 149

Getty

Im

ag

es, Inc.

Infield / Outfield Grass Line: 95-ft radius from front of pitching rubber

Infield Second base First base 13-ft radius Third base 13-ft radius Home plate 13-ft radius Foul line Foul line 90 ft between bases Pitching mound 9-ft radius

For Exercises 81–84, refer to the following: ■ C H A L L E N G E

81. What is the area enclosed in the circular sector with radius 95 feet and central angle Round to the nearest hundred square feet.

82. Approximate the area of the infield by adding the area in blue to the result in Exercise 81. Neglect the area near first and third bases and the foul line. Round to the nearest hundred square feet.

83. If a batter wants to bunt a ball so that it is fair (in front of home plate and between the foul lines) but keep it in the dirt (in the sector in front of home plate), within how small of an area is the batter trying to keep his bunt? Round to the nearest square foot.

84. Most bunts would fall within the blue triangle in the diagram on the left. Assume the catcher only fields bunts that fall in the sector described in Exercise 83 and the pitcher only fields bunts that fall on the pitcher’s mound. Approximately how much area do the first baseman and third baseman eachneed to cover? Round to the nearest square foot.

150°? In Exercises 75–78, determine whether each statement is true or false.

79. If a smaller gear has radius and a larger gear has radius and the smaller gear rotates what is the degree measure of the angle the larger gear rotates?

80. If a circle with radius has an arc length associated with a particular central angle, write the formula for the area of the sector of the circle formed by that central angle, in terms of the radius and arc length.

s1 r1 u°1 r₂ r1 ■_{C O N C E P T U A L}

In Exercises 73 and 74, explain the mistake that is made. 73. A circle with radius 5 centimeters has an arc that is made

from a central angle with measure Approximate the arc length to the nearest millimeter.

Solution:

Write the formula for arc length. Substitute and

into the formula. Simplify.

This is incorrect. What mistake was made?

s325 cm s(5 cm) (65) u65° r5 centimeters sru 65°.

74. For a circle with radius centimeters, find the area of the circular sector with central angle measuring Round the answer to three significant digits. Solution:

Write the formula for area of a circular sector.

Substitute and into the formula.

Simplify.

This is incorrect. What mistake was made? A60.5 cm2 u25° r2.2 centimeters A1 2r 2_u r u25°. r2.2 ■ C A T C H T H E M I S T A K E A1 2(2.2 cm) 2_(25°)

75. The length of an arc with central angle in a unit circle is 45.

76. The length of an arc with central angle in a unit circle is .

77. If the radius of a circle doubles, then the arc length (associated with a fixed central angle) doubles.

78. If the radius of a circle doubles, then the area of the sector (associated with a fixed central angle) doubles.

p 3

p 3 45°

You may think that a baseball field is a circular sector but it is not. If it were, the distances from home plate to left field, center field, and right field would all be the same (the radius). Where the infield dirt meets the outfield grass and along the fence in the outfield are arc lengths associated with a circle of radius 95 feet and with a vertex located at the pitcher’s mound (not home plate).

In the chapter opener about a Ford Expedition with standard

-inch rims, we learned that

the onboard computer that determines distance (odometer reading) and speed (speedometer)

combines the number of tire rotations and the size of the tire. Because the onboard

computer is set for

-inch rims (which corresponds to a tire with

-inch diameter),

if the owner decided to upgrade to

-inch rims (corresponding to a tire with

-inch

diameter), the computer would have to be updated with this new information. If the

computer is not updated with the new tire size, both the odometer and speedometer

readings will be incorrect.

You will see in this section that the

angular speed

(rotations of tires per second),

radius

(of

the tires), and

linear speed

(speed of the automobile) are related. In the context of a circle, we

will first define

linear speed

, then

angular speed

, and then relate them using the

radius

.

Linear Speed

It is important to note that although velocity

and speed

are often used as synonyms, speed

is how fast you are traveling, whereas velocity is the speed in which you are

traveling

and

the direction you are traveling. In physics the difference between speed and

velocity is that velocity has direction and is written as a vector (Chapter 7), and speed is

the

magnitude

of the velocity vector, which results in a real number. In this chapter,

speed

will be used.

Recall the relationship between distance, rate, and time:

Rate is speed, and in

words this formula can be rewritten as

It is important to note that we assume speed is constant. If we think of a car driving around

a circular track, the distance it travels is the arc length

s

, and if we let

v

represent speed

and

t

represent time, we have the formula for speed around a circle (

linear speed

):

s

v

s

t

distance

speed

ⴢ

time or speed

distance

time

d

rt

.

28.2

19

25.7

17

17

Linear Speed

D

E F I N I T I O N

If a point

P

moves along the circumference of a circle at a constant speed, then the

linear speed

v

is given by

where

s

is the arc length and

t

is the time.

v

s

t

C O N C E P T U A L O B J E C T I V E

■

Relate angular speed to linear speed.

L I N E A R A N D A N G U L A R S P E E D S

S E C T I O N

3.3

S K I L L S O B J E C T I V E S

■

Calculate linear speed.

■

Calculate angular speed.

■

Solve application problems involving angular and

linear speeds.

151 c03.qxd 8/22/11 7:08 PM Page 151

E X A M P L E 1

Linear Speed

A car travels at a constant speed around a circular track with circumference equal to 2 miles. If the car records a time of 15 minutes for 9 laps, what is the linear speed of the car in miles per hour?

Solution:

Calculate the distance traveled around the circular track.

Substitute and into

Convert the linear speed from miles per minute to miles per hour. Simplify.

■Y O U R T U R N A car travels at a constant speed around a circular track with

circumference equal to 3 miles. If the car records a time of 12 minutes for 7 laps, what is the linear speed of the car in miles per hour?

Angular Speed

To calculate linear speed, we find how fast a position along the circumference of a circle is

changing. To calculate angular speed, we find how fast the central angle is changing.

v72 mph va18 mi 15 minba 60 min 1 hr b v 18 mi 15 min vs t. s18 miles t15 minutes s(9 laps)a2 mi lap b18 mi ■Answer:105 mph

If a point

P

moves along the circumference of a circle at a constant speed, then the

central angle

␪

that is formed with the terminal side passing through point

P

also

changes over some time

t

at a constant speed. The angular speed

␻

(omega) is

given by

where

␪

is given in radians

v

u

t

Angular Speed

D

E F I N I T I O N

E X A M P L E 2

Angular Speed

A lighthouse in the middle of a channel rotates its light in a circular motion with constant speed. If the beacon of light completes one rotation every 10 seconds, what is the angular speed of the beacon in radians per minute?

Solution:

Calculate the angle measure in radians associated with one rotation.

Substitute and into vu t. v 2p (rad) 10 sec t10 seconds u2p u2p s Classroom Example 3.3.1*

A car travels at a constant speed around a circular track with circumference equal to 1.5 miles. How many laps would the car need to complete in 20 minutes in order to average a linear speed of 75 miles per hour? Answer:1623laps

Classroom Example 3.3.2 A lighthouse in the middle of a channel rotates its light in a circular motion with constant speed. If the beacon of light completes three rotations every 12 seconds, find its angular speed in radians per minute. Answer:30prad/min Study Tip

The units of angular speed will be in radians per unit time (e.g., radians per minute).

Convert the angular speed from radians per second to radians per minute.

Simplify.

■Y O U R T U R N If the lighthouse in Example 2 is adjusted so that the beacon rotates one time every 40 seconds, what is the angular speed of the beacon in radians per minute?

v12p rad/min v2p (rad)

10 sec ⴢ 60 sec

1 min

3.3 Linear and Angular Speeds 153

■Answer:v3p rad/min

If a point

P

moves at a constant speed along the

circumference of a circle with radius

r

, then the

linear speed

v

and the angular speed

are

related by

or

Note:

This relationship is true only when is

given in radians.

u

v

v

r

v

r

v

v

R

E L AT I N G L I N E A R A N D A N G U L A R S P E E D S x y r P s ␪ Study Tip

This relationship between linear speed and angular speed assumes the angle is given in radians.

Relationship Between Linear

and Angular Speeds

In the chapter opener, we discussed the Ford Expedition with

-inch standard rims that

would have odometer and speedometer errors if the owner decided to upgrade to

-inch

rims without updating the onboard computer. That is because

angular speed

(rotations of

tires per second),

radius

(of the tires), and

linear speed

(speed of the automobile) are

related. To see how, let us start with the definition of arc length (Section 3.2), which comes

from the definition of radian measure (Section 3.1).

WOR DS MATH

Write the definition of radian measure.

Write the definition of arc length

in radians).

Divide both sides by

t

.

Rewrite the right side of the equation.

Recall the definitions of

linear

and

angular

speeds.

and

␻

Substitute and

␻

into

s

v

r

v

t

r

u

t

.

ⴝ

␪

t

v

ⴝ

s

t

ⴝ

␪

t

v

ⴝ

s

t

s

t

r

u

t

s

t

r

u

t

s

r

u

(

u

u

s

r

19

17

c03.qxd 8/22/11 7:08 PM Page 153

We now will investigate the Ford Expedition scenario with upgraded tires. Notice that

tires of two different radii with the same angular speed have different linear speeds since

. The larger tire (larger

r

) has the faster linear speed.

v

r

v

12.85 in.

14.1 in.

E X A M P L E 3

Relating Linear and Angular Speeds

A Ford F-150 comes standard with tires that have a diameter of 25.7 inches. If the owner decided to upgrade to tires with a diameter of 28.2 inches without having the onboard computer updated, how fast will the truck actuallybe traveling when the speedometer reads 75 miles per hour?

Solution:

The computer in the F-150 “thinks” the tires are 25.7 inches in diameter and knows the angular speed. Use the programmed tire diameter and speedometer reading to calculate the angular speed. Then use that angular speed and the upgraded tire diameter to get the actual speed (linear speed).

STEP 1 Calculate the angular speed of the tires.

Write the formula for the angular speed. Substitute miles per hour and

into the formula.

Simplify.

STEP 2 Calculate the actual linear speed of the truck. Write the linear speed formula.

Substitute

and radians per hour.

Simplify.

Although the speedometer indicates a speed of 75 miles per hour, the actual speed is approximately 82 miles per hour .

■Y O U R T U R N Suppose the owner of the F-150 in Example 3 decides to downsize the tires from their original 25.7-inch diameter to a 24.4-inch diameter. If the speedometer indicates a speed of 65 miles per hour, what is the actual speed of the truck?

v_⬇82.296 mi hr v⬇5,214,251 in. hr ⴢ 1 mi 63,360 in. 1 mile5280 feet63,360 inches.

v⬇5,214,251 in. hr v(14.1 in.)a369,805 rad hrb v⬇369,805 r28.2 2 14.1 inches vrv v⬇369,805 rad hr v75(63,360) in./hr 12.85 in. 1 mile5280 feet63,360 inches.

r25.7 2 12.85 inches v75 vv r ■Answer:Approximately 62 mph v75 mi/hr 12.85 in. Study Tip

We could have solved Example 3 the following way: ⬇82.296 mph x28.2 in. 25.7 in.75 mph x 28.2 in. 75 mph 25.7 in.

3.3 Linear and Angular Speeds 155

In Exercises 1–10, find the linear speed of a point that moves with constant speed in a circular motion if the point travels along the circle of arc length sin time t. Label your answer with correct units.

■ _{S K I L L S} 1. 2. 3. 4. 5. (nanometers), 6. (microns), 7. 8. 9. , sec 10. ,

In Exercises 11–20, find the distance traveled (arc length) of a point that moves with constant speed valong a circle in time t.

11. 12.

13. 14.

15. 16.

17. 18.

19. , 20. ,

In Exercises 21–32, find the angular speed associated with rotating a central angle ␪in time t.

21. 22. 23. 24.

25. , hr 26. , hr 27. 28.

29. u780°,t3 min 30. u420°,t6 min 31. u900°, t3.5sec 32. u350°, t5.6sec t0.2 sec u60°, t5 sec u200°, t30.45 u18.3 t12 u7p 2 t 1 10 min up 2, t5 min u100p, t1 6sec u3p 4, t10 sec u25p, t20 min v46 km/hr t3 min v23 ft/s t27 min v120 ft/sec, t4 days v750 km/min, t10 min v72 km/hr, t15 min v60 mi/hr, t2 min v5.6 ft/sec, t20 min v4.5 mi/hr, t4.5 hr v6.2 km/hr, t3.5 sec v2.8 m/sec, t3.4 min s12.2 mm t5.2 s103 m t8 hr s2₅ cm, t4 min s₁₆1 in., t9 ns (nanoseconds) s3.6 ␮m t0.25 ms (milliseconds) s1.75 nm t12 days s7,524 mi, t250 hr s68,000 km, t3 min s12 ft, t5 sec s2 m,

E X E R C I S E S

S E C T I O N

3.3

Linear and angular speeds associated with circular motion are related through the radius rof the circle.

or

It is important to note that these formulas hold true only when angular speed is given in radians per unit of time.

vv_r vrv

S U M M A R Y

In this section, circular motion was defined in terms of linear speed (speed along the circumference of a circle) v and angular speed (speed of angle rotation)

Linear speed:

Angular speed:vu_t, where is given in radians.u vs_t

v.

S E C T I O N

3.3

33. 34.

35. 36.

37. 38.

39. , 40. ,

41. , 42. ,

In Exercises 43–52, find the distance a point travels along a circle s, over a time t, given the angular speed ␻, and radius of the circler. Round to three significant digits.

43. 44.

45. 46.

47. sec 48.

49. 50.

51. rotations per second, (express distance in miles*)

52. rotations per second, (express distance in miles*)

*1 mi5280 ft t10 min v6 r17 in., t15 min v5 r15 in., r5 cm, v5p rad 3 sec , t9 min r30 cm, v p rad 10 sec, t25 sec r6.5 cm, v2p rad 15 sec, t50.5 min t100 v3p rad 2 sec , r12 m, t3 min vp rad 4 sec, r3.2 ft, t10 min v p rad 15 sec, r5.2 in., t11 sec v6prad sec, r2 mm, t10 sec vp rad 6 sec, r5 cm, r22.6 mm v27.3 rad sec r40 cm v10p rad sec r10.2 in. v p rad 8 min r7 3 yd v16p rad 3 sec r4.5 cm v8p rad 15 sec, r2.5 in. v4p rad 15 sec, r24 ft v5p rad 16 sec, r5 mm v p 20 rad sec, r8 cm v3p rad 4 sec , r9 in. v2p rad 3 sec ,

In Exercises 33–42, find the linear speed of a point traveling at a constant speed along the circumference of a circle with radius rand angular speed ␻.

53. Tires.A car owner decides to upgrade from tires with a diameter of 24.3 inches to tires with a diameter of 26.1 inches. If she doesn’t update the onboard computer, how fast will she actually be traveling when the speedometer reads 65 mph?

54. Tires.A car owner decides to upgrade from tires with a diameter of 24.8 inches to tires with a diameter of 27.0 inches. If she doesn’t update the onboard computer, how fast will she actually be traveling when the speedometer reads 70 mph?

55. Planets.The Earth rotates every 24 hours (actually 23 hours, 56 minutes, and 4 seconds) and has a diameter of 7926 miles. If you’re standing on the equator, how fast are you traveling in miles per hour (how fast is the Earth spinning)? Compute this using 24 hours and then with 23 hou