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0.1: Angles and Radian Measure
© 2008 Roy L. Gover (www.mrgover.com)
Learning Goals:
•Define radians as another angle measure.
•Review angle measure
Calculus Prerequisite Knowledge
Definition
Initial Side
Terminal Side
Vertex Angle A
is in standard position
A x
y
Definition
A
If the terminal side moves counter-clockwise, angle A is positive
x
y
Definition
A
If the terminal side moves counter-clockwise, angle A is positive
x
y
Definition
A
If the terminal side moves counter-clockwise, angle A is positive
x
y
Definition
A
If the terminal side moves clockwise, angle A is negative
x
y
Definition
A
If the terminal side moves clockwise, angle A is negative
x
y
Definition
A
If the terminal side moves clockwise, angle A is negative
x
y
Definition
A
If the terminal side moves clockwise, angle A is negative
x
y
Definition
A
If the terminal side is on an axis, angle A is a quadrantel angle
x
y
Definition
A
If the terminal side is on an axis, angle A is a quadrantel angle
x
y
Definition
A
If the terminal side is on an axis, angle A is a quadrantel angle
x
y
Definition
A
If the terminal side is on an axis, angle A is a quadrantel angle
x
y
Is this an angle?
Try ThisWhat 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 ThisWhat is the measure of this quadrantal angle?
x
y
0°
Try ThisIf a 143° angle is in standard position, determine the quadrant in which the terminal side lies.
x
y
2
Try ThisIf a 280° angle is in standard position, determine the quadrant in which the terminal side lies.
x
y
4
Important IdeaThere are two units of measure for angles:•degrees: used in geometry•radians: used in calculusIn Precal, we use
degrees and radians.
-1 1
-1
1
y
x
Radian: The length of the arc above the angle divided by the radius of the circle.
Definition
sr
s
r , in radians
(rads)
-1 1
-1
1
y
x
Definition
s
1
s , in radians (rads)
Unit Circle: the circle with radius of 1 unitIf r=1, =s
1
DefinitionThe 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
ExampleFind the degree and radian measure of the angle in standard position formed by rotating the
terminal side ½of a circle in the positive direction. Leave your radian answer in terms of .
ExampleFind the degree and radian measure of the angle in standard position formed by rotating the
terminal side 5/6of a circle in the negative direction. Leave your radian answer in terms of .
Try ThisFind the degree and radian measure of the angle in standard position formed by rotating the
terminal side 2/3of a circle in the positive direction. Leave your radian answer in terms of .
Solution
2360 240
3
2 42
3 3
radians
Example
rads
360°
rads
6.28
2
Example
45 (degrees)
4
radians
.785 radians
Example
90 (degrees)
2
radians
1.57 radians
Try This
rads
180
rads
3.14
Try This
rads
-180
rads--
3.14
Try This
rads
270
rads
4.71
1 31 or
2 2
Do you see a pattern?
Important IdeaRadian 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 IdeaIf a circle contains 360° or 2 radians, how many radians are in 180°
• Use to change rads to degrees
180° rad
s• Use to change
degrees to rads rad
s180°
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
Change radians to
degree measure.
3
4
Example
Change 2.356 radians to degree measure.
(hint: radians are not always stated in terms of .)
Try This
Change radians to
degree measure.
157.5°
7
8
Try This
Change -3.5 radians to degree measure to the nearest tenth.
-200.5°
Definition
0
2
32
The quadrantal angles in radians 2
Definition
0
2
32
The quadrantal angles in radians 2
Definition
0
2
32
The quadrantal angles in radians 2
Definition
0
2
The quadrantal angles in radians 2
The terminal side is on an axis.
DefinitionCoterminal Angles: Angles that have the same terminal side.
Important IdeaIn precal, angles can be larger than 360° or 2 radians.
Example
Find positive angles and negative angles that are coterminal with 30°.
Important IdeaTo 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.
Analysis30° and 390° have the same terminal side, therefore, the angles are coterminal
30°x
y
x
y390°
Analysis30° and 750° have the same terminal side, therefore, the angles are coterminal
30°x
y
x
y750°
Analysis30° and 1110° have the same terminal side, therefore, the angles are coterminal
30°x
y
x
y 1110°
Analysis30° and -330° have the same terminal side, therefore, the angles are coterminal
30°x
y
x
y -330°
How many angles can you find that are coterminal with a 30° angle and how do you find them?
With Mrs. Love
Try ThisFind 3 angles coterminal with 60°
420°,780° and -300°
Try ThisFind one positive angle and two negative angle coterminal with radians.
32
and 72
2
52
,
Try ThisFind two positive angle and one negative angle coterminal with radians.
56
and 76
196
176
,
Lesson CloseIn your own words and without looking at your notes, write a definition for:•Coterminal angle
•Radian
Practice
1.cws0-1/1-10 (slides 1-37)
2. cws/11-27 (slides 38-63)