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] $458.366{}^\circ$

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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.

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}