Angles and Their
Measure
1.1
Objectives




Students will be able
Students will be able
Students will be able
Students will be able
real-life problems.
to
to
to
to
describe angles.
use radian measure.
us degree measure.
use angles to model and solve
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
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.
Coterminal Angles
An angle of xº is coterminal with angles of
xº + k · 360º
where k is an integer.
The initial and terminal sides are the same.
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.
The circumference is
2πr units
See Pg. 135

r
O
r
Finding Complements and
Supplements
• Angles are complementary if their sum is
π/2.
• Angles are supplementary if their sum is π.
• Pg. 142 #18a, 20a, 22a, 24, 30, 32a, 34a
•
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
Conversion between Degrees
and Radians
•
•
•
•
•
Using the basic relationship
 radians = 180º
2π radians = 360º
= (one complete rotation)
To convert degrees to radians, multiply
degrees by ( radians) / 180
To convert radians to degrees, multiply radians
by 180 / ( radians)
Example
Convert each angle in degrees to
radians
40º
75º
-160º
Example cont.
Solution:
2
40  40 

180 9



5
75  75 

180 12



 8  8 180
o


 160
9
9  rads
o
Length of a Circular Arc
Let r be the radius of a circle and  the nonnegative radian measure of a central angle
of the circle. The length of the arc
intercepted by the central angle is
s
s=r

Pg. 144 # 90, 96, 98
O
r
Area of a Sector of a Circle
For a circle of radius r, the area A of a sector of the
circle with central angle  (in radians) is given by
1 2
A r 
2
s

Pg. 144 #102
r
Angular Speed
Consider a particle moving at a constant
speed along a circular arc of radius r. If
s is the length of the arc travelled in
time t, then
arch length
s

Linear Speed  
time
t
If Ѳ is the angle measure corresponding to
arc length s then
central angle 
Angular Speed


time
t
• Groups: Pg 143 – 145 # 18b, 22b,
32b, 44, 56, 70, 76, 92, 104, 112
• Homework: 17 – 33 odd, 41 – 79 odd,
89 – 103 odd, 111