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Radian Measure


  • Radians is the ratio of arc length \((l)\) to the radius \((r)\) of any circle is:

\[ \theta = \frac{{\color{Green}l}}{{\color{purple}r}} \]

  • For a circle of radius \(r\), the angle \(\theta\) will be one radian the corresponding arc length is of length \(r\). This means that
    \[1 \mbox{ radian} \approx 57.2058 \mbox{ degrees}\]

  • Radians are a unit-less quantity.
  • One complete revolution in radians. 

\begin{eqnarray*} \theta &=& \frac{\text{Circumference of the circle}}{\text{radius}} \\ &=& \frac{2\pi {\cancel{r}}}{\cancel{r}} \\ &=& 2\pi \end{eqnarray*}

  • Hence \(360^{\circ}=2\pi\) or \(180^{\circ} =\pi\).
  • In calculus, (and other branches of mathematics) angles are measured entirely in radians.

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