Radian Measure
(3.1)
JMerrill, 2007
Revised 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.
The central angle is an
angle that has its vertex
at the center of a circle
r
r
θ
r
The Radian
1 radian ≈ 57.3o
4 radians ≈ 229.2o
2 radians ≈ 114.6o
5 radians ≈ 286.5o
3 radians ≈ 171.9o
6 radians ≈ 343.8o
Conversion Factor Between
Radians and Degrees
 180  180r
r d :r 


  
1radian  ?deg rees
 180  180
o
1


57.296




  
  d

d  r :d 

 180  180
1deg ree  ? radians
 
1 
    0.0017radians 
 180 
Radians can be expressed in
decimal form or exact answers.
The majority of the time, answers
will be exact--left in terms of pi
You Do

196o = ? Radians (exact answer)
  196 49

196 


45
 180  180

1.35 radians = ? degrees
 180  243
o
1.35 
 77.3


  
Arc Length


In geometry, an arc length is represented
by “s”
If any of these parts are unknown, use
the formula
s  r
Where theta is in radians
s
r
θ
r
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
  4

o
o
convert 240 to radians. 240  240   
 180 
s  r
 4
 4
 3
 16
 16.76inches

3

3
Things You MUST Remember:
s  r




π radians = 180 degrees ( ½ revolution)
2π radians = 360 degrees (1 revolution)
¼ revolution = ? degrees = ? radians
90 degrees π/2 radians
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

2
π
0o
360o
180o
3
2
2
Special Angles & The Unit Circle
P130
Evaluating Trig Functions for
Angles Using Radian Measure


Evaluate sin in exact terms
3

 is equivalent to what degree?
3

So

3
sin  sin 60 
3
2
o
60o
You Do

Evaluate cos


6
in exact terms
3
cos  cos 30 
6
2
o
Recall: Reference Angles
Reference 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
5
4
5
6

3

4

6
Using Reference Angles
a) sin 330° =
d) cos
= - sin 30°
= - 1/2
b) cos 0° =
5

4
  sin

  cos

=1
c) sin
7

6
2
2

4
3
2

6
Using Reference Angles
e) cos
5

3
 cos
1

2

3