Question

Find the radian measure corresponding to the following degree measure:
\[-{{47}^{\circ }}30'\]

Answer 1

Hint: For the above question we will have to know about the relation between degree and radian which is as follows:
\[{{1}^{\circ }}=\dfrac{\pi }{180}radians\]
Also, we will have to convert the minute (‘) into degree by the following conversion:
\[1'={{\dfrac{1}{60}}^{\circ }}\]

Complete step-by-step answer:
We have been given the angle equal to \[-{{47}^{\circ }}30'\].
So first of all we will convert 30’ into degree and as we know that \[1'={{\dfrac{1}{60}}^{\circ }}\].
\[\Rightarrow 30'={{\left( \dfrac{1}{60}\times 30 \right)}^{\circ }}={{\dfrac{1}{2}}^{\circ }}\]
We also know that \[{{1}^{\circ }}=\dfrac{\pi }{180}radians\]
\[\begin{align}
  & \Rightarrow {{\dfrac{1}{2}}^{\circ }}=\dfrac{\pi }{180}\times \dfrac{1}{2}radians \\
 & \Rightarrow \dfrac{\pi }{360}radians \\
\end{align}\]
Also,
\[\begin{align}
  & \Rightarrow {{47}^{\circ }}=\dfrac{\pi }{180}\times 47radians \\
 & \Rightarrow \dfrac{47\pi }{180}radians \\
\end{align}\]
So, \[-{{47}^{\circ }}30'=-\left( \dfrac{47\pi }{180}+\dfrac{\pi }{360} \right)radians\]
On taking LCM of 180 and 360, we get as follows:
\[-{{47}^{\circ }}30'=-\left( \dfrac{2\times \left( 47\pi \right)+\pi }{360} \right)radians=-\left( \dfrac{94\pi +\pi }{360} \right)radians=\dfrac{-95\pi }{360}radians=\dfrac{-19\pi }{72}radians\]
Therefore, the radians value of \[-{{47}^{\circ }}30'\] is equal to \[\dfrac{-19\pi }{72}radians\].

Note: Be careful at the final answer and don’t miss the negative sign before \[-{{47}^{\circ }}30'\] anywhere. Also, be careful while converting the degrees into radians and don’t use \[{{1}^{\circ }}=\dfrac{180}{\pi }radians\]. Don’t forget to change the minute in degree than into radians.