Angles and Their
Measure
Section 4.1
Objectives
• I can label the unit circle for radian angles
• I can draw and angle showing correct
rotation in Standard Format
• I can calculate Reference Angles
• I can determine what quadrant an angle is
in
Angles
An angle is formed by two rays that have a
common endpoint called the vertex. One ray is
called the initial side and the other the terminal
side. The arrow near the vertex shows the
direction and the amount of rotation from the initial
side to the terminal side.
C
A
è
Initial Side
Terminal Side
B
Vertex
Section 4.1: Figure 4.2, Angle in
Standard Position
Section 4.1: Figure 4.3, Positive
and Negative Angles
Angles of the Rectangular Coordinate System
An angle is in standard position if
• its vertex is at the origin of a rectangular coordinate system and
• its initial side lies along the positive x-axis.
y
y
 is positive
Terminal
Side

Vertex
Initial Side
x
Vertex
Initial Side
x
Terminal
Side

 is negative
Positive angles rotate counterclockwise.
Negative angles rotate clockwise.
Measuring Angles Using Degrees
The figures below show angles classified by
their degree measurement. An acute angle measures
less than 90º. A right angle, one quarter of a complete
rotation, measures 90º and can be identified by a small
square at the vertex. An obtuse angle measures more
than 90º but less than 180º. A straight angle has
measure 180º.

Acute angle
0º <  < 90º
90º
Right angle
1/4 rotation

Obtuse angle
90º <  < 180º
180º
Straight angle
1/2 rotation
Definition of a Radian
• One radian is the measure of the central
angle of a circle that intercepts an arc equal
in length to the radius of the circle.
Radian Measure
Consider an arc of length s on a circle or radius r.
The measure of the central angle that intercepts
the arc is
s
 = s/r radians.

r
O
r
Section 4.1: Figure 4.6,
Illustration of Six Radian
Lengths
Section 4.1: Figure 4.7, Common
Radian Angles
Definition of a Reference Angle
• Let  be a non-acute angle in standard
position that lies in a quadrant. Its
reference angle is the positive acute angle
´ prime formed by the terminal side or 
and the x-axis.
Example
Find the reference angle , for the following
b
angle:
 =315º
Solution:
315
a
 =360º - 315º = 45º
a
45
b
P(a, b)
Example
Find the reference angles for:
345
360  345  15
 135
 135  360  225  180  45
5
6
5 6 5 




6
6
6
6
11
4
11
3
3 
 2 
 

4
4
4
4
Homework
• WS 8-1
• Start learning Unit circle