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)