lecture 1.4 angles & radian

8/7/2019 Lecture 1.4 Angles & Radian

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0.1: Angles and RadianMeasure

© 2008 Roy L. Gover (www.mrgover.com)

Learning Goals:

Define radians as anotherangle measure.

Review angle measure

Calculus Prerequisite Knowledge



Definition

Initial Side

Terminal Side

Vertex

Angle Ais instandard

position

A x

y



Definition

A

If theterminal

sidemovescounter-clockwise,angle A is

positive

x

y



Definition

A

If theterminal


positive

x

y



Definition

A

If theterminal

sidemovesclockwise,angle A isnegative

x

y



Definition

A

If theterminal


x

y



Definition

A

If theterminal

side is onan axis,angle A is aquadrantelangle

x

y



Definition

A

If theterminal


x

y



Definition

A

If theterminal


x

y

Is this an angle?



Try This

What kind of angle is this?(hint: whereis theterminal

side?)

quadrantal angle



Try This What is the

measure of this angle?

a. -90°b. -45°

c. -270°

d. -360°





a. -90°b. -45°

c. -270°

d. -360°




measure of thisangle?a. 0°

b. 45°

c. 90°d. 120°

e. 180°





a. -90°b. -45°

c. -270°

d. -360°



Try This

What is themeasure of thisquadrantalangle? x

y

0°



Try ThisIf a 143°angle is in

standard

position,determine

the quadrant in which theterminal side

lies.

x

y

2



Important Idea

There are two units of measure for angles:

degrees: used in geometryradians: used in calculus

In Precal, we use degreesand radians.



-1 1

-1

1

y

x

Radian: Thelength of thearc above the

angle dividedby the radius of

the circle.

Definition

sr

U

s

r

U ! , U in radians (rads)



-1 1

-1

1

y

x

Definition

s

U

1

s

U ! , U in radians (rads)

Unit Circle:the circle withradius of 1

unit If r =1, = sU

1



Definition

The radian measure of anangle is the distance traveled

around the unit circle. Sincecircumference of a circle is

2 r and r =1, the distancearound the unit circle is 2T T



Ex ample

Find the degreeand radianmeasure of the

angle in standardposition formed by

rotating the terminal side ½of a circle in the positivedirection. Leave your radian

answer in terms of .T



Ex ample



rotating the terminal side 5/6of a circle in the negativedirection. Leave your radian




Try This



rotating the terminal side 2/3of a circle in the positivedirection. Leave your radian




Solution

2360 240

3r ! rg

2 42

3 3!g radians



Ex ample

} rads

360°

rads

6.282T



Ex ample

45

(degrees)

4

T

radians} .785 radians



Ex ample

90

(degrees)

2

T

radians} 1.57 radians



Try This

} rads

180

radsT

3.14



Try This

} rads

-180

rads-T

-3.14



Important Idea

Radian measure allows theexpansion of trig functions

to model real-worldphenomena whereindependent variables

represent distance or timeand not just an angle

measure in degrees.



Important Idea

If a circle contains 360° or2T radians, how manyradians are in 180°

Use to change

rads to degrees

180°

T rads Use to change

degrees to rads

T rads

180°



Ex ample

Change 30° to radianmeasure in terms of T .



Try This


2 rads3

T



Try This


4 rads3

T



Ex ample

Change radians to

degree measure.

3

4

T



Ex ample

Change 2.356 radians to

degree measure.

(hint: radians are not always stated in terms of

T .)



Try This

Change radians to

degree measure.

157.5°

7

8

T



Try This

Change -3.5 radians to

degree measure to thenearest tenth.

-200.5°



Definition

0

2

T

T

3

2

T

The

quadrantalangles in

radians

2T



Definition

0

2

T

T

The

quadrantalangles in

radians

2T

The terminal side is on an

axis.



DefinitionCoterminal Angles: Anglesthat have the same terminal

side.Important Idea

In precal, angles can belarger than 360° or 2

radians.

T



Ex ample

Find positive angles andnegative angles that arecoterminal with 30°.



Important Idea

To find coterminal angles,simply add or subtract

either 360° or 2 radiansto the given angle or any

angle that is alreadycoterminal to the givenangle.

T



y



Analysis

30° and1110° havethe same

terminalside,therefore,the anglesare

coterminal

30° x

y

x

y1110°

y



Analysis

30° and-330° havethe same

terminalside,therefore,the anglesare

coterminal

30° x

y

x

y-330°



How many angles can youfind that are coterminal

with a 30° angle and howdo you find them?



Try This

Find 3 angles coterminalwith 60°

420°,780° and -300°



Try ThisFind one positive angle andtwo negative angle

coterminal with

radians.

32

T

and72

T

2T

52

T ,



Try ThisFind two positive angle andone negative angle

coterminal with

radians.

56

T

and76

T 196

T 176

T ,



Lesson CloseIn your own words andwithout looking at your

notes, write a definition for:

Coterminal angle

Radian

lecture 1.4 angles & radian

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