Radian Measure (3.1)

JMerrill, 2007Revised 2000

A Newer Kind of Angle Measurement: The Radian 1 radian = the measure of the

central angle of a circle that intercepts an arc equal in length to the radius of the circle.

r

r r

θ

The central angle is an angle that has its vertex at the center of a circle

The Radian

1 radian ≈ 57.3o 2 radians ≈ 114.6o 3 radians ≈ 171.9o

4 radians ≈ 229.2o 5 radians ≈ 286.5o 6 radians ≈ 343.8o

Conversion Factor Between Radians and Degrees

:180 180

dd r d

180 180:

rr d r

1 ?deg

180 1801 57.296o

radian rees

Radians can be expressed in decimal form or exact answers. The majority of the time, answers will be exact--left in terms of pi

1deg ?

1 0.0017180

ree radians

radians

You Do 196o = ? Radians (exact answer)

1.35 radians = ? degrees

196 49196

180 180 45

180 2431.35 77.3o

Arc Length In geometry, an arc length is

represented by “s” If any of these parts are unknown,

use the formula

r

r sθs r

Where theta is in radians

Arc Length Example: A circle has a radius of 4

inches. Find the length of the arc intercepted by a central angle of 240o.

We will use s = rθ, but first we have to convert 240o to radians.

4240 240

180 3o

4 16 4 16.76

3 3

s r

inches

Things You MUST Remember: π radians = 180 degrees ( ½

revolution) 2π radians = 360 degrees (1

revolution) ¼ revolution = ? degrees = ? radians 90 degrees π/2 radians

s r

Exact Angle Measurement Angle measures that can be

expressed evenly in degrees cannot be expressed evenly in radians, and vice versa. So, we use fractional multiples of π.

Quadrant angles

0o180oπ

32

2

2

360o

Special Angles & The Unit Circle

P130

Evaluating Trig Functions for Angles Using Radian Measure Evaluate in exact terms

is equivalent to what degree?

So

sin3

3

60o

sin sin 6023

3o

You Do Evaluate in exact terms

cos6

cos cos3026

3o

Recall: Reference AnglesReference Angle: the smallest positive acute angle determined by the x-axis and the terminal side of θ

ref angle ref angle

ref angle ref angle

Find Reference Angle

150°

30°

225°

45°

300°

60°

5

3

3

5

4

4

5

6

6

Using Reference Anglesa) sin 330° =

= - sin 30°

= - 1/2

b) cos 0° =

= 1

5c) sin

4

sin4

2

2

7d) cos

6

cos6

3

2

Using Reference Angles5

e) cos3

cos3

1

2