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Radians Prepared by Title V Staff: Daniel Judge, Instructor Ken Saita, Program Specialist East Los Angeles College EXIT BACKNEXT © 2002 East Los Angeles College. All rights reserved. Click one of the buttons below or press the enter key

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A radian is defined as the length of a circular arc that is equal to the radius of the circle. EXIT BACKNEXT

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If we choose a circle with a radius of 1, the circle is referred to as a unit circle. If a central angle of the unit circle intercepts an arc with length 1 then the measure of the central angle is 1 radian. So, we can use arc length to measure angles. EXIT BACKNEXT

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Arc Length on a Unit Circle If you walked along the arc of this circle, one complete revolution, you would have walked a distance of 2 units. EXIT BACKNEXT

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Why? For any circle of radius r: c = 2 r so in a circle with radius 1 c = 2 (1) = 2 EXIT BACKNEXT

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How far would one travel if they walked half way around the unit circle? EXIT BACKNEXT

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How far would one travel if they walked one-quarter the distance around the unit circle? EXIT BACKNEXT

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What does the distance traveled along the arc of a unit circle have to do with radians? EXIT BACKNEXT

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The radian system of measurement is based on using arc length: is the distance traveled along the arc of a unit circle. EXIT BACKNEXT

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This means that if the central angle measures 90 , this angle is equivalent to radians. To illustrate... EXIT BACKNEXT

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90 radians ¼ revolution ccw EXIT BACKNEXT

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180 radians ½ revolution ccw EXIT BACKNEXT

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270 radians ¾ revolution ccw EXIT BACKNEXT

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360 2 radians one revolution ccw EXIT BACKNEXT

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If one were to complete two revolutions ccw, what would this measure in radians? EXIT BACKNEXT

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360 + 360 = 720 which is equivalent to 2 + 2 = 4 2 n gives the radian measurement for n complete counter-clockwise revolutions. EXIT BACKNEXT

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What if we walked along the arc of a unit circle clockwise instead of counterclockwise? We simply insert a negative sign to indicate movement in the clockwise direction, as with degrees. EXIT BACKNEXT

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Pictures to remember Revolve ccwRevolve cw m( ) = sm( ) = -s s = measure of arc length EXIT BACKNEXT

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End of Radians Title V East Los Angeles College 1301 Avenida Cesar Chavez Monterey Park, CA 91754 Phone: (323) 265-8784 Email Us At: menteprog@hotmail.com Our Website: http://www.matematicamente.org EXIT BACKNEXT

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