Conversion of Radians into Degrees and Vice Versa

We know that the circumference of a circle of radius r is 2\pi r = \left( l \right) and the angle formed by one complete revolution is \theta radian; therefore,


radian-to-degree

\begin{gathered} \theta = \frac{1}{r} \\ \Rightarrow \theta = \frac{{2\pi r}}{r} \\ \Rightarrow \theta = 2\pi \,radian \\ \end{gathered}

Thus we have the relationship

\begin{gathered} 2\pi \,radian\, = 360^\circ \\ \Rightarrow \pi \,radian = 180^\circ \\ \Rightarrow 1{\text{ }}radian = \frac{{180^\circ }}{\pi }\,\,\, = \frac{{180^\circ }}{{3.1416}}\,\,\,\, = \,\,\,57.296^\circ \\ \end{gathered}

Further

\begin{gathered} 1^\circ = \frac{\pi }{{180}}\,radian \\ 1^\circ = \frac{{3.1416}}{{180}}\,\,\,\,\, = 0.0175\,radian \\ \end{gathered}

Example:

Convert the following angles into degrees:

(i) \frac{{2\pi }}{3} radians
(ii) 3 radians

Solution:

(i)

\begin{gathered} \frac{{2\pi }}{3}\,radians = \frac{2}{3}\left( {\pi \,radian} \right) \\ \frac{{2\pi }}{3}\,radians = \frac{2}{3}\left( {180^\circ } \right)\,\,\,\, = 120^\circ \\ \end{gathered}

(ii)

3{\text{ }}radians\; = 3{\text{ }}\left( {1{\text{ }}radian} \right) = {\text{ }}3\left( {57.296^\circ } \right)\,\,\,\, = 171.888^\circ

Example:

Convert 54^\circ \,\,45' into radians.

Solution:

\begin{gathered} 54^\circ \,\,45' = \left( {54\frac{{45}}{{60}}} \right)^\circ = \left( {54\frac{3}{4}} \right)^\circ \,\,\, = \frac{{219^\circ }}{4} \\ \Rightarrow 54^\circ \,\,45' = \frac{{219}}{4}\left( {1^\circ } \right) \\ \Rightarrow 54^\circ \,\,45' = \frac{{219}}{4}\left( {0.0175} \right) = 54^\circ \,\,45' = 0.958\,radians \\ \end{gathered}

Most calculators automatically would convert degrees into radians and radians into degrees.

Example:

An arc subtends an angle of 70^\circ at the center of a circle and its length is 132mm. Find the radius of the circle.

Solution:

70^\circ \,\,\, = 70 \times \frac{\pi }{{180}} = \frac{{70}}{{180}}\left( {\frac{{22}}{7}} \right) = \frac{{11}}{9}\,\,\,\,\,\,\,\,\,\,\left( {\because \pi = \frac{{22}}{7}} \right)
\therefore \theta = \frac{{11}}{9}\,radian and l = 132m.m
\therefore \theta = \frac{1}{r} \Rightarrow r = \frac{l}{\theta } = 132 \times \frac{9}{{11}} = 108m.m

Example:

Find the length of the equatorial arc subtending an angle of 1^\circ at the center of the Earth, taking the radius of the Earth as 5400km.

Solution:

1^\circ = \frac{\pi }{{180}} = \frac{{3.1416}}{{180}}\,radians

\therefore \theta = \frac{{3.1416}}{{180}} and r = 6400\,km
Now
\theta = \frac{l}{r}
 \Rightarrow l = r\theta = 6400 \times \frac{{31416}}{{1800000}} = 111.7\,km