3Radian Measureand the Unit CircleApproach

How does an odometer orspeedometer on an automobilework? 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.

CourtesyFord M

otor Company

*Section 3.3, Example 3 and Exercises 53 and 54.

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I N TH IS CHAPTE R, you will learn a second way to measure angles using radians. You will convert betweendegrees and radians. You will calculate arc lengths, areas of circular sectors, and angular and linear speeds. Finally, the thirddefinition of trigonometric functions using the unit circle approach will be given. You will work with the trigonometricfunctions in the context of a unit circle.

129129

Arc Length Area of a Circular

Sector

TrigonometricFunctions and theUnit Circle (CircularFunctions)

Properties of CircularFunctions

The Radian Measureof an Angle

Converting BetweenDegrees and Radians

Linear Speed Angular Speed Relationship

Between Linear andAngular Speeds

3.1Radian Measure

3.3Linear and

Angular Speeds

3.4Definition 3 ofTrigonometricFunctions: Unit Circle Approach

3.2Arc 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).

RADIAN MEASURE AND THEUNIT CIRCLE APPROACH

L E A R N I N G O B J E C T I V E S

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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 writetrigonometric 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 hasits vertex at the center of a circle.

When the intercepted arcs length is equal to the radius, the measure of the central angleis 1 radian. From geometry, we know that the ratio of the measures of two angles is equalto the ratio of the lengths of the arcs subtended by those angles (along the same circle).

u1

u2

s1s2

360.

r

= 1 radian

r

r

Note that both s and r are measured in units of length. When both are given in the sameunits, the units cancel, giving the number of radians as a dimensionless (unitless) realnumber.

C A U T I O N

To correctly calculate radians from

the formula the radius and

arc length must be expressed in thesame units.

u sr ,

CONCE PTUAL OBJ ECTIVES

Understand that degrees and radians are both measures of angles.

Realize that radian measure allows us to writetrigonometric functions as functions of real numbers.

RADIAN M EASU R ESECTION

3.1

SKI LLS OBJ ECTIVES

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, Thisleads to a general definition of radian measure.

s1 r.u1 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 MeasureD E F I N I T I O N

s

r

r

130

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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.

ufull rotation 2pr

r 2p

2p2pr

Answer: 0.3 rad

EXAM PLE 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 measureto arc length and radius.

Let and

YOUR TURN What is the measure (in radians) of a central angle that interceptsan arc of length 3 inches on a circle with radius 50 inches?

u 4 ft

10 ft 0.4 radr 10 feet.s 4 feet

u s

r

u

Answer: 0.06 rad

EXAM PLE 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 TA K E

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

COR R ECT

Write the formula relating radianmeasure to arc length and radius.

Substitute andinto the radian expression.

Convert the radius (2) meters tocentimeters:

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

u 0.03 rad

u 6 cm

200 cm

2 meters 200 centimeters

u 6 cm

2 m

r 2 meterss 6 centimeters

u (in radians) s

r

INCOR R ECT

Substitute andinto the radian expression.

ERROR (not converting both numeratorand denominator to the same units)

3

u 6 cm

2 m

r 2 meterss 6 centimeters

YOUR TURN What is the measure (in radians) of a central angle that interceptsan 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 mustbe the same in order to use

u s

r

Study Tip

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

u

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

Answer: rad12

u

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

Answer: rad32

u

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Because radians are unitless, the word radians (or rad) is often omitted. If an anglemeasure is given simply as a real number, then radians are implied.

WORDS 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 betweenthem. We start by considering one full rotation around the circle. An angle correspondingto one full rotation is said to have measure , and we saw previously that one fullrotation corresponds to rad.

WORDS 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

180p

180 p rad

360 2p rad

u 2pr

r 2p radu (in radians)

s

r.s 2pr

s 2pr

u 360

u 2p360

u 4u

u 4u

132 CHAPTER 3 Radian Measure and the Unit Circle Approach

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.

urud

aunit conversations, like 1 hr60 min

b

To convert degrees to radians, multiply the degree measure by

ur ud a p180b

p

180 .

CONVERTING DEGREES TO RADIANS

To convert radians to degrees, multiply the radian measure by

ud ur a180p b

180p

.

CONVERTING RADIANS TO DEGREES

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Before we begin converting between degrees and radians, lets first get a feel forradians. How many degrees is 1 radian?

WORDS 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). Lets compare two

angle