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Geometric Representation of Angles
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Angles
Initial Side and Standard Position
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Degrees: One degree is 1/360 of a revolution.
A right angle is an angle that measures 90 degrees or ¼ revolution
A straight angle is an angle that measures 180 degrees or ½ revolution
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Drawing an Angle
(a) 45 degrees
(b) -90 degrees
(c) 225 degrees
(d) 405 degrees
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1 degree equals 60’ (minutes)
1’ (minute) equals 60” (seconds)
Using graphing calculator to convert
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Definition
Arc Length For a circle of radius r, a central angle of
radians subtends an arc whose length s is
s=r
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Find the length of the arc of a circle of radius 2 meters subtended by a central angle of 0.25 radian.
s=rwith r = 2 meters and Θ = 0.25
2(0.25) = 0.25 meter
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One revolution is 2π therefore, 2πr = rθ (arc length formula)
It follows then that 2π = θ and
1 revolution = 2π radians 360 degrees = 2π radians or 180 degrees = π radians so . . . 1 degree = π/180 radian and 1 radian = 180/π degrees
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Convert each angle in degrees to radians:
(a) 60 degrees (b) 150 degrees (c) – 45 degrees (d) 90 degrees
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Convert each angle in radians to degrees
(a) π/6 radian (b) 3π/2 radian (c) -3π/4 (d) 7π/3
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Page 375 has common angles in degree and radian measures
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Steps: (1) Find the measure of the central angle
between the two cities (2) Convert angle to radians (3) Find the arc length (remember we live
on a sphere and the distance between two cities on the same latitude is actually an arc length)
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The area A of the sector of a circle of radius r formed by a central angle of θ radians is
A = ½ r^2θ
Examples
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Linear Speed:
v = s/t
Angular Speed:
ω = θ/t
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Angular Speed is usually measured in revolutions per minute (rpms).
Converting to radians per minute
Linear Speed given an Angular Speed:
v = rω where r is the radius
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A child is spinning a rock at the end of a 2-ft rope at the rate of 180 rpms. Find the linear speed of the rock when it is released.
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At the Cable Car Museum you can see four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, caused by rotating wheel whose diameter is 8.5 feet. How fast is the wheel rotating? Express your answer in rpms.
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On-line Examples
On-line Tutorial