Slide 1-1

3Radian Measure and the Unit Circle

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3.1 Radian Measure

3.2 Applications of Radian Measure

3.3 The Unit Circle and Circular Functions

3.4 Linear and Angular Speed

Radian Measure and the Unit Circle

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3.1 Radian Measureπr2 circle a of nceCircumfere

0360 revolution complete One

length arcs

radius

r

πr2s when 2

.

.

.

2rs when radians 2

rsn radian whe 1

:Definition

degrees 360 radians 2

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Example 1: Convert each degree measure to radians

000 7.325 )( 135 )( 108 )( cba

Example 2: Convert each radian measure to degrees.

92.2 )( 6

7 )(

12

11 )( cba

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Example 3: Find each function value.

3

5sec )(

4

3cot )(

6

5cos )(

cba

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3.2 Applications of Radian Measure

πr2s when 2

.

.

.

2rs when radians 2

rsn radian whe 1

:Definition

Arc length formula

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2circle complete a of Area πr

22

22sector a of Area rπr

π

θ

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Examples 1-4: Pages 101-102

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3.3 The Unit Circle and Circular Functions

Slide 1-10

Example:

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3.4 Linear and Angular Speed

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Linear Speed

Suppose that a point P moves at a constant speed along a circle of radius r and center O.

The measure of how fast the position of P is changing is the linear speed. If v represents linear speed, then

or

where s is the length of the arc traced by point P at time t.

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Angular Speed

As point P moves along the circle, ray OP rotates about the origin.

The measure of how fast angle POB is changing is its angular speed.

is the angular speed, θ is the measure of angle POB (in radians) at time t.

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Slide 1-15

Example FINDING LINEAR SPEED AND DISTANCE TRAVELED BY A SATELLITE

A satellite traveling in a circular orbit 1600 km above the surface of Earth takes 2 hr to make an orbit. The radius of Earth is approximately 6400 km.

(a) Approximate the linear speed of the satellite in kilometers per hour.

The distance of the satellite from the center of Earth is approximately r = 1600 + 6400 = 8000 km.

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For one orbit, θ = 2π, and

Since it takes 2 hours to complete an orbit, the linear speed is

(b) Approximate the distance the satellite travels in 4.5 hr.