Calculus Prerequisite Knowledge
0.1: Angles and Radian
Measure
Learning Goals:
•Review angle measure
•Define radians as another
angle measure.
© 2008 Roy L. Gover (www.mrgover.com)
Definition
Terminal Side
y
Vertex
A
Initial Side
x
Angle A
is in
standard
position
Definition
y
A
x
If the
terminal
side
moves
counterclockwise,
angle A is
positive
Definition
y
A
x
If the
terminal
side
moves
counterclockwise,
angle A is
positive
Definition
y
A
x
If the
terminal
side
moves
counterclockwise,
angle A is
positive
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
x
If the
terminal
side
moves
clockwise,
angle A is
negative
Definition
y
A
If the
terminal
side is on
x an axis,
angle A is a
quadrantel
angle
Definition
y
A
If the
terminal
side is on
x an axis,
angle A is a
quadrantel
angle
Definition
y
A
If the
terminal
side is on
x an axis,
angle A is a
quadrantel
angle
Definition
y
If the
terminal
side is on
A
x an axis,
angle A is a
quadrantel
Is this an angle? angle
Try This
What kind of
angle is this?
(hint: where
is the
terminal
side?)
quadrantal angle
Try This
What is the
measure of
this angle?
a. -90°
b. -45°
c. -270°
d. -360°
Try This
What is the
measure of
this angle?
a. -90°
b. -45°
c. -270°
d. -360°
Try This
What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°
Try This
What is the
measure of this
angle?
a. 0°
b. 45°
c. 90°
d. 120°
e. 180°
Try This
What is the
measure of
this angle?
a. -90°
b. -45°
c. -270°
d. -360°
Try This
What is the
measure of
this angle?
a. -90°
b. -45°
c. -270°
d. -360°
Try This
What is the
measure of
this
quadrantal
angle?
0°
y
x
Try
This
If a 143°
y
angle is in
standard
position,
determine
the quadrant
in which the
terminal side
lies.
x
2
Try
This
If a 280°
y
angle is in
standard
position,
determine
the quadrant
in which the
terminal side
lies.
x
4
Important Idea
There are two units of
measure for angles:
•degrees: used in geometry
•radians: used in calculus
In Precal, we use degrees
and radians.
Definition y
Radian: The
length of the
arc above the
angle divided
by the radius of
the circle.
1
r

-1
s
1
x
-1
s
  ,  in radians (rads)
r
Definition y
Unit Circle:
the circle with
radius of 1
unit
If r=1,  =s
1
1

-1
s
1
x
-1
s
  ,  in radians (rads)
1
Definition
The radian measure of an
angle is the distance traveled
around the unit circle. Since
circumference of a circle is
2  r and r=1, the distance
around the unit circle is 2 
Example
Find the degree
and radian
measure of the
angle in standard
position formed by
rotating the terminal side ½
of a circle in the positive
direction. Leave your radian
answer in terms of  .
Example
Find the degree
and radian
measure of the
angle in standard
position formed by
rotating the terminal side 5/6
of a circle in the negative
direction. Leave your radian
answer in terms of  .
Try This
Find the degree
and radian
measure of the
angle in standard
position formed by
rotating the terminal side 2/3
of a circle in the positive
direction. Leave your radian
answer in terms of  .
Solution
2
360  240
3
2
4
2 
radians
3
3
Example
360°
2
rads
 6.28 rads
Example
45 (degrees)

radians
4
 .785 radians
Example
90 (degrees)

radians
2
 1.57 radians
Try This
180 

 3.14
rads
rads
Try This

-180
-
rads
 -3.14 rads
Try This
Do you see a pattern?
270 
1
3
1  or
2
2
rads
 4.71 rads
Important Idea
Radian measure allows the
expansion of trig functions
to model real-world
phenomena where
independent variables
represent distance or time
and not just an angle
measure in degrees.
Important Idea
If a circle contains 360° or
2 radians, how many
radians are in 180°
Use to change
rads to degrees
Use
to
change
rads

•
180° degrees to rads
180°
•
 rads
Example
Change 30° to radian
measure in terms of .
Try This
Change 120° to radian
measure in terms of .
2
rads
3
Try This
Change 240° to radian
measure in terms of .
4
rads
3
Example
3
Change
radians to
4
degree measure.
Example
Change 2.356 radians to
degree measure.
(hint: radians are not
always stated in terms of
.)
Try This
7

Change
radians to
8
degree measure.
157.5°
Try This
Change -3.5 radians to
degree measure to the
nearest tenth.
-200.5°
Definition 
The
quadrantal
angles in
radians
2

0
2
3
2
Definition 
The
quadrantal
angles in
radians
2

0
2
3
2
Definition 
The
quadrantal
angles in
radians
2

0
2
3
2
Definition 
2
The
quadrantal
0
angles in

2
radians
The terminal side is on an
axis.
Definition
Coterminal Angles: Angles
that have the same terminal
side.
Important Idea
In precal, angles can be
larger than 360° or 2 
radians.
Example
Find positive angles and
negative angles that are
coterminal with 30°.
Important Idea
To find coterminal angles,
simply add or subtract
either 360° or 2  radians
to the given angle or any
angle that is already
coterminal to the given
angle.
Analysis
30° and
390° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
390°
x
Analysis
30° and
750° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
750°
x
Analysis
30° and
1110° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
1110°
x
Analysis
30° and
-330° have
the same
terminal
side,
therefore,
the angles
are
coterminal
y
30°
x
y
-330°
x
With Mrs. Love
How many angles can you
find that are coterminal
with a 30° angle and how
do you find them?
Try This
Find 3 angles coterminal
with 60°
420°,780° and -300°
Try This
Find one positive angle and
two negative angle
3

coterminal with
2
radians.

7
5


and

,
2
2
2
Try This
Find two positive angle and
one negative angle
coterminal with  5
6
radians.
19
7
17

and

,
6
6
6
Lesson Close
In your own words and
without looking at your
notes, write a definition for:
•Coterminal angle
•Radian
Practice
1.cws0-1/1-10
2. cws/11-27
(slides 1-37)
(slides 38-63)