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 ${{1}^{\circ }}=\dfrac{\pi }{180}radians$

& \Rightarrow {{\dfrac{1}{2}}^{\circ }}=\dfrac{\pi }{180}\times \dfrac{1}{2}radians \\\ & \Rightarrow \dfrac{\pi }{360}radians \\\ \end{aligned}$$ Also, $$\begin{aligned} & \Rightarrow {{47}^{\circ }}=\dfrac{\pi }{180}\times 47radians \\\ & \Rightarrow \dfrac{47\pi }{180}radians \\\ \end{aligned}$$ So, $$-{{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.