lecture 1.3.1 angles & radian

8/7/2019 Lecture 1.3.1 Angles & Radian

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0.1: Angles and RadianMeasure

Learning Goals:

Define radians as anotherangle measure.

Review angle measure

Calculus Prerequisite Knowledge


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Definition

Initial Side

Terminal Side

Vertex

Angle Ais instandard

position

A x

y


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Definition

A

If theterminal

sidemovescounter-clockwise,angle A is

positive

x

y


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Definition

A

If theterminal


positive

x

y


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Definition

A

If theterminal


positive

x

y


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Definition

A

If theterminal

sidemovesclockwise,angle A isnegative

x

y


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Definition

A

If theterminal


x

y


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Definition

A

If theterminal


x

y


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Definition

A

If theterminal


x

y


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Definition

A

If theterminal

side is onan axis,angle A is aquadrantelangle

x

y


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Definition

A

If theterminal


x

y


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Definition

A

If theterminal


x

y


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Definition

A

If theterminal


x

y

Is this an angle?


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Try This

What kind ofangle is this?(hint: whereis theterminal

side?)

quadrantal angle


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Try This What is the

measure ofthis angle?

a. -90b. -45

c. -270

d. -360


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a. -90b. -45

c. -270

d. -360


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measure of thisangle?a. 0

b. 45

c. 90d. 120

e. 180


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measure of thisangle?a. 0

b. 45

c. 90d. 120

e. 180


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a. -90b. -45

c. -270

d. -360


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a. -90b. -45

c. -270

d. -360


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Try This

What is themeasure ofthisquadrantalangle? x

y

0


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Try ThisIf a 143angle is in

standard

position,determine

the quadrantin which theterminal side

lies.

x

y

2


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Try ThisIf a 280angle is in

standard

position,determine

the quadrantin which theterminal side

lies.

x

y

4


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Important Idea

There are two units ofmeasure for angles:

degrees: used in geometryradians: used in calculus

In Precal, we use degreesand radians.


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-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)


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-1 1

-1

1

y

x

Definition

s

U

1

s

U ! , U in radians (rads)

Unit Circle:the circle withradius of 1

unitIfr=1, =sU

1


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Definition

The radian measure of anangle is the distance traveled

around the unit circle. Sincecircumference of a circle is

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


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Example

Find the degreeand radianmeasure of the

angle in standardposition formed by

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

answer in terms of .T


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Example



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



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Try This



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



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Solution

2360 240

3r ! rg

2 42

3 3!g radians

X

X


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Example

} rads

360

rads

6.282T


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Example

45

(degrees)

4

T

radians} .785 radians


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Example

90

(degrees)

2

T

radians} 1.57 radians


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Try This

} rads

180

radsT

3.14


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Try This

} rads

-180

rads-T

-3.14


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Try This

} rads

270

rads

4.71

1 3

1 or2 2

T

T

Do you see a pattern?


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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.


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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


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Example

Change 30 to radianmeasure in terms ofT.


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Try This


2 rads3

T


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Try This


4 rads3

T


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Example

Change radians to

degree measure.

3

4

T


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Example

Change 2.356 radians to

degree measure.

(hint: radians are notalways stated in terms of

T.)


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Try This

Change radians to

degree measure.

157.5

7

8

T


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Try This

Change -3.5 radians to

degree measure to thenearest tenth.

-200.5


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Definition

0

2

T

T

3

2

T

The

quadrantalangles in

radians

2T


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Definition

0

2

T

T

3

2

T

The

quadrantalangles in

radians

2T


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Definition

0

2

T

T

3

2

T

The

quadrantalangles in

radians

2T


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Definition

0

2

T

T

The

quadrantalangles in

radians

2T

The terminal side is on an

axis.


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DefinitionCoterminal Angles:Anglesthat have the same terminal

side.Important Idea

In precal, angles can belarger than 360 or 2

radians.

T


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Example

Find positive angles andnegative angles that arecoterminal with 30.


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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


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Analysis

30 and390 havethe same

terminalside,therefore,the anglesare

coterminal

30x

y

x

y

390

y


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Analysis



coterminal

30x

y

x

y

750

y


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Analysis



coterminal

30x

y

x

y1110

y


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Analysis

30 and-330 havethe same


coterminal

30x

y

x

y-330


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How many angles can youfind that are coterminal

with a 30 angle and howdo you find them?


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Try This

Find 3 angles coterminalwith 60

420,780 and -300


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Try ThisFind one positive angle andtwo negative angle

coterminal with

radians.

32

T

and72

T

2T

52

T,


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Try ThisFind two positive angle andone negative angle

coterminal with

radians.

56

T

and76

T 196

T 176

T,


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Lesson CloseIn your own words andwithout looking at your

notes, write a definition for:

Coterminal angle

Radian

lecture 1.3.1 angles & radian

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