Section 4.1
Radian and Degree
Measures
Angles in Pre-Calculus
Vertex
90°
y
y
180°
Initial
side
Standard
Position
O Initial side
270°
x
x
0°
360°
Definition of Coterminal Angles
Coterminal angles are angles with the
same initial side and terminal side.
90° (-270°)
Initial side
y
Counterclockwise
positive angle
A
180°
0°
O
(-180°)
360°
B
(-360°)
Angle A and angle B
Clockwise
are coterminal
negative angle
angles.
270° (-90°)
x
Example 1
Find one positive and one negative
coterminal angle for a 30° angle.
HINT:
one rotation counterclockwise = 360°
one rotation clockwise = -360°
Positive coterminal angle = 30°+ 360° = 390°
y
30°
x
Negative coterminal angle = 30°- 360° = -330°
In what quadrant is a 140°? Find one
positive and one negative coterminal
angle for a 140° angle.
2nd quadrant
140° + 360° = 500°
140° - 360° = -220°
Another measure of angles is called
radians.
Radians are the angle measures used
in calculus.
Radians
A radian is the measure of a central
angle θ that intercepts an arc “s” equal
in length to the radius “r” of the circle.
y
r
Θ = 1 rad.
r
s=r
x
360° = 2π radians
360
2π
1 radian 
1 
2π
360
π
180
1 
1 radian 
180
π
Formula to go
Formula to go
from degrees to from radians to
radians.
degrees.
In radians,
1 rotation counterclockwise = 2π
1 rotation clockwise = -2π
π
90°
rad
2
y
π rad
180°
00°rad
360°
2π
rad
x
3π
rad
270°
2
Example 2
Find one positive and one negative
4π
coterminal angle for
.
3
In what quadrant is this angle?
3rd quadrant
One positive
4π 6π 10π
4π


 2π 
3
3
3
3
One negative
4π
4π 6π
2π
 2π 


3
3
3
3
Example 3
Change the following angle measures
to radians.
a. 60°
b. -50°
a.
b.
60
π
π
60 

180
3
50
π
5π
50 

180
18
Example 4
Change the following angle measures
to degrees.
a.
b.
π
4
11π

6
a.
π
4
π 180

 45
4 π
b.
11π

6
11π 180


 330
6
π
Example 5
Start at the terminal side of the
given angle Θ in standard position.
Find the radian measure of the
resulting angle, in standard position,
after the given number of rotations.
Give answers in terms of π.
a.
π
2
rad, 1 counterclockwise
2
3
π 5
 2π 
2 3
π 10π

2
3
3π 20π 23π


6
6
6
Example 6
Start at the terminal side of the
given angle Θ in standard position.
Find the degree measure of the
resulting angle, in standard position,
after the given number of rotations.
a.
1
20, clockwise
3
1
20   360 
3
20  120  100