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

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”.