Circular Functions

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Circular Functions

• Radian Measure
• Text 3.2 Radians and Degrees
• 3.4 Arc Length and Area of a Sector
• 3.5 Velocities

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Objectives

• To Define Radian Measure using central angles of
  rotation.
• Convert from Degree to Radian measurement.
• Applications of Radian Measure
• Angular Distance
• Linear Distance (Arc Length)
• Angular and Linear Velocities

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Warm-Up

1. Rotating Light The red light on the top of a
   police car rotates through one complete
   revolution every 2 seconds. Through how many
   degrees does it rotate in 1 second?
2. Rotating Light A searchlight rotates through
   one complete revolution every 4 seconds. How long
   does it take the light to rotate 90?
3. Clock Through how many degrees does the hour
   hand of a clock move in 4 hours?
4. Rotation of the Earth It takes the earth 24
   hours to make one complete revolution on its
   axis. Through how many degrees does the earth
   turn in 12 hours?

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Additional Examples Pg 128 95The Campagnolo
Hyperon carbon wheel has 22 spokes evenly
distributed around the rim of the wheel. What is
the measure, in radians, of the central angle
formed by adjacent pairs of spokes?
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Formulas

• Angular Distance
• Linear Distance (Arc Length)
• Area of a Sector

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Example 1 pg 119A central angle O in a circle of
radius 3 cm cuts off an arc length of 6 cm. What
is the radian measure of O ?
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Example 1 pg 141Give the length of the arc cut
off by the central angle of 2 radians in a circle
of radius 4.2 inches.
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The Area of a SectorA ½ r²O
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Applications

• Angular Distance
• Linear Distance (Arc Length)

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In navigation, distance is not usually measured
along a straight line, but along a great circle
because the Earth is round.
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Additional Examples pg 125 7Angle Between
Cities Los Angeles and San Francisco are
approximately 450 miles apart on the surface of
the earth. Assuming that the radius of the earth
is 4000 miles, find the radian measure of the
central angle with its vertex at the center of
the earth that has Los Angeles on one side and
San Francisco on the other side.
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Additional Example Pg 126 21 and 22If a
central angle with its vertex at the center of
the earth has a measurement of 1, then the arc
on the surface of the earth that is cut off by
this angle (known as the great circle distance)
has a measure of 1 nautical mile.(Note a
regular mile is a statute mile.)
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Example 2 Pg 141Below is a model of a Ferris
wheel with diameter 250 ft, and O is a central
angle formed as a rider travels from his initial
position P0 to position P1. Find the distance
traveled by the rider if O 45 and if O 105.
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Example 3 pg 141
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Example 4 Pg 143
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Velocities
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