Relation between Degrees and Radians

The radian measure of an angle is the ratio of the length of the arc (s) whose centre is at the vertex of the angle to the radius (r).

$\displaystyle \begin{array}{{|l|}}\hline \displaystyle \theta =\frac{s}{r} \\ \hline \end{array}$

For one complete anticlockwise revolution, the length of the arc is the length of entire circumference and hence $ \displaystyle s=2\pi r$. Therefore, the radian measure of one complete revolution is

$\displaystyle \begin{array}{{|l|}}\hline \displaystyle \theta =\frac{2\pi r}{r}{}=2\pi \ \text{radians} \\ \hline \end{array}$  

In degrees, one complete counterclockwise revolution is $\displaystyle 360°$ and hence $ \displaystyle \theta=360°$.

$ \displaystyle \begin{array}{l}\therefore \ \ 360{}^\circ =2\pi \ \text{radians}\\\\\therefore \ \ 180{}^\circ =\pi \ \text{radians}\\\\\therefore \ \ 1{}^\circ \times 180=\pi \ \text{radians}\\\end{array}$

$\displaystyle \therefore\ $ $\displaystyle \begin{array}{{|l|}}\hline \displaystyle 1{}^\circ =\frac{\pi }{{180}}\ \text{radians} \\ \hline \end{array}$

$ \displaystyle \begin{array}{l}\ \ \ \ \text{Since }\pi \ \text{radians}=180{}^\circ ,\\\\\ \ \ \ 1\ \text{radian}\ \times \pi =180{}^\circ \end{array}$

$\displaystyle \therefore\ $$\displaystyle \begin{array}{{|l|}}\hline \displaystyle 1\ \text{radian}\ =\frac{{180{}^\circ }}{\pi } \\ \hline \end{array}$

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