4.1
Radian and Degree
Measure
Initial side
Coterminal Angles Angles that differ by multiples
of 360 degrees or 2 radians.
Find a positive and negative coterminal angle
for each of the following.
13

a.
6
3
b.
4
13
13 12
 2 

6
6
6
11 5
,
4
4
25   11

, ,
6 6
6
What is a radian?
3 radians
2 radians

1 radian
r=1
Complementary Angles – two angles are complementary
if their sum is 90 degrees or  .
2
Supplementary Angles have a sum of 180 degrees or .
Find the complementary and supplementary angles for
2
 2
5 4 
5




2
5
10
10
10
2
5 2 3




5
5
5
5
Conversions:
Radians
Degrees
To convert degrees to radians, multiply by
 rad
180


180
To convert radians to degrees, multiply by
 rad

Converting an angle from D M ' S " to decimal form.

152 15'29" 


 15   29 
152     
 
 60   3600


152.25806
Arc Length
s  r
s = arc length
r = radius
theta = radian measure
s
  60

r = 4 in.
First, we need to convert degrees to radians.
4
s  4 
inches  4.1888 inches
3
3

Area of a Sector of a Circle
1 2
A r 
2
where
 is measured in radians
A sprinkler on a golf course fairway is set to spray water over a distance
of 70 feet and rotates through an angle of 120 degrees. Find the area of the
fairway watered by the sprinkler.


120o = how many radians?
2
120 
radians
3
120
70 ft

1
2 2  4900
2
A  70  
 5131ft
 3 
2
3
79-99 odd, 107
Linear and Angular Speeds
Consider a particle moving at a constant speed along a circular arc of
radius r. If s is the length of the arc traveled in time t, then the linear
speed v of the particle is
Linear speed v

arc length s


time
t
Moreover, if
is the angle (in radian measure) corresponding to the arc
length s, then the angular speed
(the lowercase Greek letter omega)
of the particle is


central angle 

Angular speed  

time
t

A relationship between linear speed and angular speed is

v  r
Finding Linear Speed
The second hand of a clock is 10.2 cm long. Find the linear speed of the
Tip of the second hand as it passes around the clock face.
Linear speed v
Arc length s
arc length s


time
t
 2r  2(10.2)  20.4 cm

s 20.4 cm
,v  
=1.068 cm/sec
t
60 sec

