O B J E C T I V ES :

1. U S E A R O TAT I N G R AY T O E X T EN D T H E D EF IN IT I O N O F A N G L E M E A S U R E T O N EG AT I V E A N G L E S AN D A N GL ES G R E AT ER T H A N 1 8 0 ° .

2. D EF IN E R A D I A N M EA S U R E A N D C O N V E RT A N GL E M EA S U R E S B E T W EE N D E GR EE S A N D R A D I A N S.

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

1803602)1(22

rC

r = 1

0° = 360° = 2π

290

180° = π

23270

Example #1Convert from Radians to Degrees

A.

B.

C.

9

53

5

20180

108180

900180

Example #2Convert from Degrees to Radians

A. 150°

B. -330°

C. 540°

65

180

611

180

3180

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.

34

37

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.

rwholepartL

ncecircumferewholepartL

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:

2rwholepartA

areawholepartA

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

A.

2

2

2

cm 25.205

3196196

31

19621

32

1423

2

rwholepartA

cm 32.293

282831

2821

32

14223

2

2

rwholepartL

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

B.

2

2

2

cm 66.125

40

144185

12360100

rwholepartA

ft 94.203

20

24185

122360100

2

rwholepartL

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

12

457

8

10

11

Example #6Arc Length

5’’

in6.232

151043

526045

2

rwholepartL