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,
\[\begin{gathered} \theta = \frac{l}{r} \\ \Rightarrow \theta = \frac{{2\pi r}}{r} \\ \Rightarrow \theta = 2\pi \,radian \\ \end{gathered} \]
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
$$\theta = \frac{l}{r}$$
$$ \Rightarrow l = r\theta = 6400 \times \frac{{31416}}{{1800000}} = 111.7\,km$$