Understanding Radians and Degrees: Conversions and Applications
Radians offer an alternative to degrees for measuring angles, based on arc length ratios. For instance, when the arc length equals the radius, the angle in radians is 1 (θ = a/r). In a semicircle, the connection between degrees and radians reveals π radians equals 180°, establishing relationships for various angles. This article discusses converting degrees to radians and vice versa using key examples and presents the concept of angular velocity in the context of musical vinyl records, illustrating practical applications of radians in everyday scenarios.
Understanding Radians and Degrees: Conversions and Applications
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Presentation Transcript
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 Here a = ½ of circumference ie = ½ of d = r r So (radians) = r /r =
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.
Converting 180 X degrees radians X 180 The fraction button on your calculator ab/c can be very useful here
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
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”.
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
