Angles and Their Measure

Section 3.1

Objectives• Convert between degrees, minutes, and

seconds (DMS) and decimal forms for angles.

• Find the arc length of a circle.• Convert from degrees to radians, and from

radians to degrees.• Find the area of a sector of a circle.• Find the linear speed of an object traveling

in circular motion.

Background Info

• Ray• Vertex• Angle

– Initial side– Terminal side

• Counterclockwise/positive rotation• Clockwise/negative rotation• Standard position

Background Info• Quadrantal angles/angles that lie in quadrant

• Measures of rotation: Degrees and Radians

Draw the following angles:

45° -90°

225° 405°

Converting between DMS & Degrees

• 1 counterclockwise revolution = 360°• 1° = 60’ (60 minutes)• 1’ = 60” (60 seconds)• Make sure calculator is set in degrees

mode• Example: Convert 21.256° to DMS:

• Example: Convert 50°6’21” to a decimal in degrees

Radians• Central angle (θ): angle whose vertex is at

the center of a circle• Measure of 1 radian:

length of radius = arc length• Find the arc length (s) of a circle using the

following formula:

s = rθ• Central angle must be in radians in order to

use this formula.• Example: Page 125 #71

Convert from Degrees to Radians and from Radians to Degrees

• Since one revolution is 360°, and the circumference of a circle equals 2πr, then

s = rθ

2πr = rθ

θ = 2π radians

and

1 revolution = 2π radians

Therefore, 180° = π radians

Convert from Degrees to Radians and from Radians to Degrees

• Degrees to radians

Multiply by

• Radians to degrees

Multiply by

180

180

Convert to radians

60°

-150°

107°

Convert to degrees

3 radians

6

34

Memorize the table on page 121

Pages 124-125 (11-77 odds)

Check answers in the back of the book

Find the Area of a Sector of a Circle

• Example: Find the area of the sector of a circle of radius 2 feet formed by an angle of 30°

A1

2r2

Find the Linear Speed of an Object Traveling in Circular Motion

v s

t

t

Linear Speed Angular Speed

Page 126 #97

Pages 125-127 (79-115 odds)

Check answers in the back of the book