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Radian Measure

Converting between degrees and radians

  • To convert from radians to degrees, find the angle as a fraction of the full circle (\(2\pi\)) then multiply by \(360^{\circ}\): 
    \[ \mbox{Number of Degrees} = \frac{\mbox{Number of radians}}{2\pi}\times 360^{\circ}\]
  • For example: convert \(\displaystyle \frac{7\pi}{8} \) radians to degrees:
    \begin{eqnarray*}  
    \mbox{Number of Degrees} &=& \frac{\mbox{Number of radians}}{2\pi}\times 360^{\circ} \\  
    &=& \frac{\frac{7\pi}{8}}{2\pi}\times 360^{\circ} \\  
    &=& 157.5^{\circ}
    \end{eqnarray*}
  • To convert degrees to radians, find the angle as a fraction of the full circle (\(360^{\circ}\)) then multiply by \(2\pi\): 
    \[ \mbox{Number of Radians} = \frac{\mbox{Number of Degrees}}{360^{\circ}}\times 2\pi\] 
  • For example: convert \(135^{\circ}\) to radians:
    \begin{eqnarray*}
    \mbox{Number of Radians} &=& \frac{\mbox{Number of Degrees}}{360^{\circ}}\times 2\pi \\
    &=& \frac{135^{\circ}}{360^{\circ}}\times 2\pi \\
    &=& \frac{3\pi}{4}
    \end{eqnarray*} 
  • Note that when possible we try to leave radians as a fraction of \(\pi\).

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