8-1 Measurement of Arcs (Presentation)

8/7/2019 8-1 Measurement of Arcs (Presentation)

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8-1 Angle Measure and Rotation

Unit 8 Trigonometric and Circular Functions


2/13

Concepts and Objectives

Arc Measurement and Rotation (Obj. #26)

Convert between degrees and radians Calculate the length of an arc intercepted by a given

angle

a cu ate t e area o a sector


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Radian Measure

Up until now, we have measured angles in degrees.

Another unit of measure that mathematicians use iscalled radian measure.

y

the center of a circle that

intercepts an arc on the

circle equal in length to the

radius of the circle has ameasure of1 radian.

r

r

x

= 1 radian


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Radian Measure

You should recall that the circumference of a circle is

given by C= 2r, where ris the radius of the circle. Anangle of 360, which corresponds to a complete circle,

intercepts an arc equal to the circumference.

= 180 radians

=

1801 radian

=1 radians

180

If no unit of angle measure is specified, then the

angle is understood to be measured in radians.


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Radians and Degrees

Converting between radians and degrees is just like

converting between any other type of units: Put the unit you are converting from on the bottom,

and the unit you are converting to on the top.

Example: Convert 120 to radians.

120120

180 180

2

3= =

1radians


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Radians and Degrees

Example: Convert 57 48' to radians

57.8

= + = 4857 48' 57 57.860

=

57.8= 0.321 1.01 radians

Example: Convert radians to degrees3

5

3 180

5( )= 3 36 = 108


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Unit Circle

=30

6

=454

=60

3

=90

2

=2

1203

= 31354

=

5150

6

=0 0 = 180

=

7210

6

= 52254

=

4240

3

=3

2702

=

5300

3

= 73154

=

11330

6

= 360 2


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Arc Length of a Circle

The length of an arc is proportional to the measure of its

central angle.

sThe length s of the arc on

r

a circle of radius rby acentral angle measuring

is given by

( must be in radians)

=s r


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Example: Find the length to the nearest hundredth of

the arc intercepted by the given central angle and radius.

= =5

1.38 ft,6

r

Rememberthe angle measure mustbe in radians. If

you are given an angle measure in degrees, you must

convert it into radians.

( )

=

1.386

s 3.61 ft


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Example: Two gears are adjusted so that the smaller

gear drives the larger one. If the smaller gear rotatesthrough an angle of 225, through how many degrees

will the larger gear rotate?25

2.5 cm 4.8 cm =225 4

= s r

=

52.5

4

=25

cm8

.

8

=125

radians192

=

125 180

192

117


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Area of a Sector of a Circle

Recall that a sectoris the portion of the interior of the

circle intercepted by a central angle. The area of thesector is proportional to the area of the circle.

r

The area A of a sector of acircle of radius rand

central angle is given by

( must be in radians)=

21

2A r


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Area of a Sector of a Circle

Example: Find the area of a sector of a circle having the

given radius r and central angle .

= =2

59.8 km,3

r

( ) =

21 259.82 3

A =2

3745 km


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Homework

College Algebra (brown book)

Page 565: 10, 15, 25-65 (5s), 68, 71, 75-90 (5s),120, 124, 126

Turn in: 30, 50, 68, 80, 88, 90, 126

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8-1 Measurement of Arcs (Presentation)