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Radians

Radians. Radian measure is an alternative to degrees and is based upon the ratio of. arc length radius. a. ie. .  (radians) = a / r. r.  - theta. If the arc length = the radius. ie. r. .  (radians) = r / r = 1. r. If we now take a semi-circle. a.

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Radians

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  1. Radians Radian measure is an alternative to degrees and is based upon the ratio of arc length radius a ie   (radians) = a/r r  - theta

  2. If the arc length = the radius ie r   (radians) = r/r = 1 r If we now take a semi-circle a Here a = ½ of circumference ie = ½ of d  = r r So  (radians) = r /r = 

  3. Since we have a semi-circle the the angle must be 180. We now get a simple connection between degrees and radians.  (radians) = 180 This now gives us 2 = 360 /2 = 90 3/2 = 270 /3 = 60 2/3 = 120 /4 = 45 3/4 = 135 /6 = 30 5/6 = 150 etc NB: radians are usually expressed as fractional multiples of.

  4. Converting 180 X  degrees radians   X 180 The fraction button on your calculator ab/c can be very useful here

  5. Ex1 72 = 72/180 X  = 2 /5 Ex2 330 = 330/180 X  = 11 /6 Ex32 /9 = 2 /9 X 180 = 2/9 X 180 = 40 Ex423 /18 = 23 /18 X 180 = 23/18 X 180 = 230

  6. Example 5 Angular Velocity In the days before CDs the most popular format for music was “vinyls”. Singles played at 45rpm while albums played at 331/3 rpm. rpm =revolutions per minute ! Going back about 70 years an earlier version of vinyls played at 78rpm. Convert these record speeds into “radians per second”.

  7. NB: 1 revolution = 360 = 2 radians 1 min = 60 secs So 45rpm = 45 X 2 or 90 radians per min = 90/60 or 3/2 radians per sec So 331/3rpm = 331/3 X 2 or 662/3  radians per min = 662/3 /60 or 10/9 radians per sec So 78rpm = 78 X 2 or 156 radians per min = 156/60 or 13/5 radians per sec

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