Angles and Their
Students will be able to describe angles.
Students will be able to use radian measure.
Students will be able to us degree measure.
Students will be able to use angles to model and solve
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.
Terminal Side Initial Side
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.
Positive angles rotate counterclockwise. Negative angles rotate clockwise.
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.
Consider an arc of length s on a circle or radius
The measure of the central angle that intercepts
the arc is
= s/r radians.
The circumference is r
2πr units O r
See Pg. 135
Finding Complements and
• Angles are complementary if their sum is
• 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
Acute angle Right angle Obtuse angle Straight angle
0º < < 90º 1/4 rotation 90º < < 180º 1/2 rotation
Conversion between Degrees
• 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)
Convert each angle in degrees to
8 8 180 o
9 9 rads
Length of a Circular Arc
Let r be the radius of a circle and the non-
negative radian measure of a central angle
of the circle. The length of the arc
intercepted by the central angle is
Pg. 144 # 90, 96, 98
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
Pg. 144 #102
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
If Ѳ is the angle measure corresponding to
arc length s then
Angular Speed central angle
• 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