Question

Find the angle measure of 4 radians
[a] $114.591{}^\circ $
[b] $141.372{}^\circ $
[c] \[229.183{}^\circ \]
[d] $282.743{}^\circ $
[e] $458.366{}^\circ $

Answer 1

Hint: Use the fact that one complete rotation is equal to $2\pi $ radians and also equal to \[360{}^\circ \]. Use the unitary method to convert 4 radians in degrees.

Complete step-by-step answer:
The angle subtended by the arc of length 1 unit at the centre of a circle of radius 1 unit is said to be equal to 1 radian. Hence in this system, one complete angle is equal to $2\pi $ radians.

We know that $2\pi $ radians are equal to \[360{}^\circ \]
Hence 1 radian is equal to $\dfrac{360{}^\circ }{2\pi }=\dfrac{180{}^\circ }{\pi }$
Hence 4 radians are equal to $\dfrac{180{}^\circ }{\pi }\times 4=229.183{}^\circ $
Hence 4 radians are equal to \[229.183{}^\circ \]
Hence option [c] is correct.

Note: The conversion can also be understood as follows.
Equal angles are subtended by equal length arcs in congruent circles.
Let 4 radians = x degrees.
Arc subtending 4 radians in a circle of radius 1 unit has length $l=1\times 4=4$. Because in the radian system $\theta =\dfrac{l}{r}$ .
Arc subtending x degrees in a circle of radius 1 unit has length $l=\dfrac{x}{360}2\pi r=\dfrac{\pi x}{180}$. Because in degree system $\theta =\dfrac{l}{2\pi r}\times 360$
Since both the lengths need to be equal, we have
 $\begin{align}
  & \dfrac{\pi x}{180}=4 \\
 & \Rightarrow x=\dfrac{4}{\pi }\times 180=229.183{}^\circ \\
\end{align}$