OBJECTIVES :

1. USE A ROTATING RAY TO EXTEND THE DEFINITION OF ANGLE MEASURE TO NEGATIVE ANGLES AND ANGLES GREATER THAN 180° .

2. DEFINE RADIAN MEASURE AND CONVERT ANGLE MEASURES BETWEEN DEGREES AND RADIANS.

6.3Angles & Radian Measure

Angles of Rotation

Positive angles are rotated counter-clockwise & negative angles clockwise.

Standard position has the initial side on the x-axis & the vertex on the origin.

Radians & the Unit Circle

Radians are used to measure angles using arc length.

Circumference:

1801or

180radian 1

180

3602)1(22

rC

r = 1

0° = 360° = 2π

290

180° = π

2

3270

Example #1Convert from Radians to Degrees

A.

B.

C.

9

5

3

5

20180

108180

900180

Example #2Convert from Degrees to Radians

A. 150°

B. -330°

C. 540°

6

5

180

6

11

180

3

180

Example #3Find the angle measures from each graph.

360° - 60°= 300°

-360° + 90° + 115° = -155°

5(180°) = 900°

Example #4Draw the following angles in standard position. State the quadrant in which the terminal side is located.

A. -110°

B. 530°

Example #4Draw the following angles in standard position. State the quadrant in which the terminal side is located.

C. 3400°

D. 3

Example #4 (continued…)Draw the following angles in standard position. State the quadrant in which the terminal side is located.

E.

F.

3

4

3

7

Arc Length of a Circle

Depending on whether an angle is given in radians or degrees the formulas for arc length vary slightly, although the concept remains the same.

For radians:

For degrees:

rL

rL

180

The key to learning this is not to memorize either formula, but to build on what you already know. The length of an arc is a fraction of the distance around the entire circle (circumference).

Multiply that fraction by the circumference of the circle and you get the arc length.

rwhole

partL

ncecircumferewhole

partL

2

Sector Area of a Circle

Depending on whether an angle is given in radians or degrees the formulas for sector area also vary.

For radians:

For degrees: 2

360rA

2

2rA

And just like arc length, the formulas for sector area are based on the same concept:

2rwhole

partA

areawhole

partA

Example #5Find the Arc Length & Sector Area of the following:

A.

2

2

2

cm 25.205

3

196196

3

1

1962

1

3

2

1423

2

rwhole

partA

cm 32.29

3

2828

3

1

282

1

3

2

14223

2

2

rwhole

partL

Example #5Find the Arc Length & Sector Area of the following:

B.

2

2

2

cm 66.125

40

14418

5

12360

100

rwhole

partA

ft 94.203

20

2418

5

122360

100

2

rwhole

partL

The second hand on a clock is 5 inches long. How far does the tip of the hand move in 45 seconds?

12

6

39

1

2

4

578

10

11

Example #6Arc Length

5’’

in6.23

2

1510

4

3

5260

45

2

rwhole

partL