4.1 Radian and Degree
measure
Definition of an angle
An angle is made from two rays with a common initial point.
Ter min al
side
Initial
side
In standard position the initial side is on the x axis
Positive angle vs. Negative angle
Positive angles are Counter clockwise
C.C.W.
Negative angles are Clockwise
C.W.
Angles with the same initial side and
terminal side are coterminal.


The measure of an angle is from
initial side to terminal side
Vertex at the origin (Center)
Central
r
Angle

r
Definition of a Radian
Radian is the measure of the arc of a unit
circle.
Unit circle is a circle with a radius of 1.
https://www.youtube.com/watch?v=HhHds http://www.youtube.com/watch?v=7QhgYX
8cRE4
QAM_Rc
Radian Protractor Worksheet
• Materials
– Worksheet
– Circle with radius
– String
– Scissors
Complete steps 1-5
The quadrants in terms of Radians
What is the circumference of a circle with
radius 1?
The quadrants in terms of Radians
What is the circumference of a circle with
radius 1?
2
1
0 2
The quadrants in terms of Radians
The circumference can be cut into parts.
We’ll start by cutting it into fourths.

2

1
3
2
0 2
The quadrants in terms of Radians
I
II

2

2
 

  
III
3
2
1
3
2
0  
0 2
3
   2
2
IV

2
• Complete steps 6-8, fill in the blanks
• Then starting with a fresh circle on the
back of your original circle, write in the
radian measurements
• Draw a fresh circle in your notes and
practice using the patterns of 4ths and
6ths.
Radian vs. Degree measurements
360º = 2
180º = 
So

1 

180
rad
or
1 rad 
180


To convert Degrees into Radians multiply by 180
To convert Radians into Degrees multiply by
180

Change 140º to Radians

140
7
140 *

    2.443460953
180 180
9
Change
7
3
to degrees
7 180 1260
*

 420
3

3
How to use radians to find Arc length
The geometry way was to find the
circumference of the circle and multiply by
the fraction. Central angle
360º
In degrees Arc length called S would be
S

360
2r 
How to use radian to find Arc length
In degrees Arc length called S would be
S

2r 
360
Rewrite the equation replacing 360 degrees
with it’s equivalent in radians, then simplify
to find the new equation for arc length

2r   S  r
S
2
S  r
Find the arc length in radians
r = 9, θ = 50º
Changing to rads


5
50 

180 18
r
Arc length S
5
S
9
18
5
S
 7.85
2
Find the Coterminal Angle
Since 2 equals 0. it can be added or
subtracted from any angle to find a
coterminal angle.
Given  3
4
 3
5
 2 
4
4
 3
 11
 2 
4
4
Linear speed and Angular speed
Linear speed is how fast a particle moves
along a circular arc.
Angular speed, is how fast the angle
changes.
• How do we calculate how fast we are
driving?
• mi/hr or the distance divided by the time
• The same idea applies to linear and
angular speed
Linear speed and Angular speed
Linear speed, v = arc length  S
time
Angular speed,
Lower case “Omega”
t
Central angle 


time
t
Assuming “constant speed”
A Ferris wheel has a 50 ft radius and makes 1.5
revolutions per minute. What is the linear and
angular speed in radians?
S r (50)(3 )
v 

 150  471.2 feet per min
t
t
1 min

3 rad
 
 3 radians
t
1 min
arclength S
v

time
t
Centralangle 


time
t
H Dub
• 4-1 Pg. 290 #1-10, 15-18, 21, 22, 55-70,
and 103