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Chapter 3Chapter 3

Radian Measure and Radian Measure and the Unit Circle the Unit Circle

ApproachApproach

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Chapter 3Chapter 3OverviewOverview

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Chapter 3Chapter 3ObjectivesObjectives

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 degree and radians.)

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Section 3.1Section 3.1Radian MeasureRadian Measure

Skills Objectives Calculate the radian

measure of an angle. Convert between degrees

and radians. Calculate trigonometric

function values for angles given in radians.

Conceptual Objectives 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.

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Radian MeasureRadian Measure

To correctly calculate radians from the

formula , the radius and arc length

must be expressed in the same units.

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Your Turn: Calculate Radian MeasureYour Turn: Calculate Radian Measure

•What is the measure (in radians) of a central

angle that intercepts an arc of length 12 mm

on a circle with radius 4 cm?

•Solution: First, convert 4 cm to 40 mm.

•Next, calculate = 0.3.40

12

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Converting Between Degrees and RadiansConverting Between Degrees and Radians

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Your Turn : Converting Between Degrees Your Turn : Converting Between Degrees and Radiansand Radians

Convert 60° to radians.

Solution: r = 60 = or 1.047.

Convert to radians.

Solution: d = = 270°.

180

3

180

2

3

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Special Angles in the Unit CircleSpecial Angles in the Unit Circle

We can now draw the unit circle with special angles in degrees and radians.

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Section 3.2 Section 3.2 Arc Length and Area of a Circular Arc Length and Area of a Circular

SectionSection

Skills Objectives 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.

Conceptual Objectives Understand that to use

the arc length formula, the angle measure must be in radians.

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Arc LengthArc Length

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Your Turn: Calculating Arc LengthYour Turn: Calculating Arc Length

In a circle with radius 15 inches, an arc is intercepted by a central angle with measure . Find the arc length.

Solution: s = 15 = 5inches.

In a circle with radius 20 m, an arc is intercepted by a central angle with measure 113°. Find the arc length. Approximate the arc length to the nearest meter.

Solution: s = ≈ 39 m

3

3

18011320

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Find the area of a slice of pizza (cut into 8 equal pieces) if the entire pizza has a 16-inch diameter.

Solution: r = 8, r = , so area = 25 in.2 .

Calculating Area of a Circular SectorCalculating Area of a Circular Sector

4

28

42

1

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Section 3.3 Section 3.3 Linear and Angular SpeedsLinear and Angular Speeds

Skills Objectives Calculate linear

speed. Calculate angular

speed. Solve application

problems involving angular and linear speeds.

Conceptual Objectives Relate angular speed to

linear speed.

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Linear and Angular SpeedLinear and Angular Speed

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Your Turn: Linear SpeedYour Turn: Linear Speed

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 in miles per hour?

Solution: t = 12 minutes, or hour and s = 7 X 3.

V = = = 105 mph.

5

1

t

s2.

21

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Relationship Between Linear and Relationship Between Linear and Angular SpeedsAngular Speeds

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Section 3.4 Section 3.4 Definition 3 of Trigonometric Functions: Definition 3 of Trigonometric Functions:

Unit Circle ApproachUnit Circle Approach

Skills Objectives Draw the unit circle

illustrating the special angles and label the sine and cosine values.

Determine the domain and range of trigonometric (circular) functions.

Classify circular functions as even or odd.

Conceptual Objectives Understand that trigonometric

functions using the unit circle approach are consistent with both of the previous definitions (right triangle trigonometry and trigonometric functions of nonacute angles in the Cartesian plane).

Relate x-coordinates and y-coordinates of points on the unit circle to the values of cosine and sine functions.

Visualize periodic properties of trigonometric (circular) functions.

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The Unit CircleThe Unit Circle

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Common Mistake: Using the Wrong Mode Common Mistake: Using the Wrong Mode on a Calculatoron a Calculator

Many calculators automatically reset to degree mode after every calculation, so be sure to always check what mode the calculator indicates.