Question

Find the radian measure corresponding to the degree $ - 47^\circ 30'$.
(A)$\dfrac{{ - 19\pi }}{{72}}rad$
(B) $\dfrac{{19\pi }}{{72}}rad$
(C) $\dfrac{{13\pi }}{{72}}rad$
(D) None of these

Answer 1

Hint: Convert $30'$ into degrees using $1' = \dfrac{1}{{60}}^\circ $ to get$ - 47^\circ 30' = - \dfrac{{95}}{2}^\circ $. Find the degree measure of $ - \dfrac{{95}}{2}^\circ $ using$1^\circ = \dfrac{\pi }{{180}}rad$ to get the answer.


Complete step by step solution:
We are given the degree $ - 47^\circ 30'$.
We need to find its radian measure.
We know that $1^\circ = \dfrac{\pi }{{180}}rad$.
We also know that $1^\circ = 60'$. This would imply that $1' = \dfrac{1}{{60}}^\circ $
We will first convert the minute part $30'$ of the degree $ - 47^\circ 30'$ into degrees.
Now, $30' = 30 \times \dfrac{1}{{60}}^\circ = \dfrac{1}{2}^\circ $.
Therefore, we have $ - 47^\circ 30' = - (47^\circ + \dfrac{1}{2}^\circ ) = - (47 + .\dfrac{1}{2})^\circ = - \dfrac{{95}}{2}^\circ $
So, to find the radian measure of the degree $ - 47^\circ 30'$, we will find the radian measure of the degree$ - \dfrac{{95}}{2}^\circ $.
Now \[1^\circ = \dfrac{\pi }{{180}}rad \Rightarrow - \dfrac{{95}}{2}^\circ = - (\dfrac{{95}}{2} \times \dfrac{\pi }{{180}})rad = - \dfrac{{19\pi }}{{72}}rad\]
Hence the radian measure corresponding to the degree $ - 47^\circ 30'$is \[ - \dfrac{{19\pi }}{{72}}rad\].
So option A is the right answer


Note: 1) To find the degree measure of a radian, we can use the formula$1rad = \dfrac{{180}}{\pi }^\circ $
2) The single apostrophe (‘) stands for minutes and the double quotation mark (“) stands for seconds.
Therefore, it is read as “47 degrees, 30 minutes, and 23 seconds”.