Obj. 17 Radian Measure (Presentation)
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Transcript of Obj. 17 Radian Measure (Presentation)
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7/30/2019 Obj. 17 Radian Measure (Presentation)
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Obj. 17 Radian Measure
Unit 5 Trigonometric and Circular Functions
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Concepts and Objectives
Arc Measurement and Rotation (Obj. #17)
Convert between degrees and radians Calculate the length of an arc intercepted by a given
angle
Calculate the area of 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.
An angle with its vertex at
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
y
= 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.
= 360 2 radians
= 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 to on the top, and
the unit you are convertingfrom on the bottom.
Example: Convert 120 to radians.
120120
180 180
2
3= =
1radians
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Radians and Degrees
Example: Convert 57 48' to radians
Example: Convert radians to degrees3
5
57.8
180
= + = 4857 48' 57 57.860
=
57.8
180= 0.321 1.01 radians
3 180
5( )= 3 36 = 108
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The Unit Circle
=0 0
=30
6
=45
4
=60
3
=90
2
=2
1203
=
31354
=
5150
6
= 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.
r
s
The length s of the arc on
a circle of radius rcreatedby a central angle
measuring is given by
( must be in radians)
=s r
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Arc Length of a Circle
Example: Find the length to the nearest hundredth of a
foot of the arc intercepted by the given central angle andradius.
Rememberthe angle measure mustbe in radians. If
you are given an angle measure in degrees, you must
convert it into radians.
= =
51.38 ft,
6r
( )
= 51.386
s = 3.61 ft
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Arc Length of a Circle
Example: Find the length to the nearest tenth of a meter
of the arc intercepted by the given central angle andradius.
= = 2.9 m, 68r
( )( )
= 2.9 68 180s
= 3.5 m
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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 areaA 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 (round to the nearestkilometer).
= =2
59.8 km,3
r
( ) =
21 259.82 3
A =2
3745 km
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Homework
College Algebra
Page 565: 25-65 (5s), 68, 75, 80, 85, 87, 88, 90, 124,126
Classwork: College Algebra Page 565: 8, 12, 14, 26, 32, 38, 43, 46, 56, 58