Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of...

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Slide 1-1 3 Radian Measure and the Unit Circle

Transcript of Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of...

Page 1: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

Slide 1-1

3Radian Measure and the Unit Circle

Page 2: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

Slide 1-2

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

Page 3: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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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

Page 4: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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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

Page 5: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

Slide 1-5

Example 3: Find each function value.

3

5sec )(

4

3cot )(

6

5cos )(

cba

Page 6: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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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

Page 7: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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2circle complete a of Area πr

22

22sector a of Area rπr

π

θ

Page 8: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

Slide 1-8

Examples 1-4: Pages 101-102

Page 9: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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3.3 The Unit Circle and Circular Functions

Page 10: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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Example:

Page 11: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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3.4 Linear and Angular Speed

Page 12: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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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.

Page 13: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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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.

Page 14: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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Page 15: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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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.

Page 16: Slide 1-1 3 Radian Measure and the Unit Circle. Slide 1-2 3.1 Radian Measure 3.2 Applications of Radian Measure 3.3 The Unit Circle and Circular Functions.

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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.


3.1 Radian Measure

3.1 Radian Measure

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3 Radian Measure and the Unit Circle Approach - Wiley

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