Section 4.1
Angles and Their Measure
ANGLES
A ray is a part of a line that has only one endpoint
and extends forever in the opposite direction. An angle
is formed by two rays that have a common endpoint. One
ray is called the initial side and the other the terminal side.
A rotating ray is often a useful way to think about angles.
The endpoint of an angle's initial side and terminal side is
the vertex of the angle.
An angle is in standard position if
1. its vertex is at the origin of a rectangular coordinate system
2. its initial side lies along the positive x-axis
Positive angles are generated by counterclockwise rotation.
Thus, angle  is positive. Negative angles are generated
by clockwise rotation as you see angle  in the diagram.
An angle is called quadrantal if its terminal side lies on the
x-axis or the y-axis. If a standard angle has a terminal side
that lies in a quadrant then we say that the angle lies in
that quadrant. Angle  lies in quadrant II. Angle  lies
in quadrant III.
Measuring Angles Using Degrees
Names of Angles
Example
What is the radian measure of  for an arc of
length 20 inches and a radius of 5 inches.
20 inches
5 inches
Relationship between Degrees and Radians
Example
Convert each angle in degrees to radians.
a. 1350
b. -1200
c. -1500
d. 900
e. 1800
Example
Convert each angle in radians to degrees.
a.

2
b. 
c. 
d.

3
5
6
2
e.
3
Drawing Angles in Standard Position
Angles Formed by Revolution of Terminal Sides
Example
Draw and label each angle in standard position.
3
a.  
2
b.  =2
7
c.  =
4
Degree and Angle Measures of
Selected Positive and Negative Angles
Example
Assume the following angles are in standard position.
Find a positive angle less than 3600 that is coterminal
with each of the following.
a. 3900
b. 4050
c. -1350
Example
Assume the following angles are in standard position.
Find a positive angle less than 2 that is coterminal
with each of the following.
a.
5
2
11
b.
4
c. -

6
Example
Find a positive angle less than 2 or 3600 that is coterminal
with each of the following.
a. 7650
b.
22
6
c. -
19
6
Example
A circle has a radius of 7 inches. Find the length
of the arc intercepted by a central angle of 1200 .
Example
A circle has a radius of 5 inches. Find the length
of the arc intercepted by a central angle of 1500 .
Example
A windmill in Holland is used to generate electricity.
Its blades are 12 feet in length. The blades rotate
at eight revolutions per minute. Find the linear
speed, in feet per minute of the tops of the blades.
Convert the angle to radian measure. 1500
(a)
(b)
(c)
(d)
2
3
5
6

6
3
4
A circle has a radius of 7 inches. Find the length of the arc
intercepted by a central angle of 210 degrees.
(a)
(b)
(c)
(d)
42
6
140
3
28
3
49
6