Question

Express the following angle into radians: $50{}^\circ 37'30''$.

Answer 1

Hint: In this problem we need to convert the given angle into radians. In the given angle we have degrees, minutes and seconds. So first we will consider the seconds and convert it into minutes by using the relation $1'=60''$. Now we will add this value to the minutes given in the question. After that we will convert obtained minutes into degrees by using the relation $1{}^\circ =60'$. Now we will get the whole given angle in degrees only. To convert the obtained degrees into radians we will use the relation $\pi =180{}^\circ $.

Complete step by step solution:
Given angle is $50{}^\circ 37'30''$.
Consider the seconds in the above given angle which is $30''$. We have the relation $1'=60''$. From this we can write
$\begin{align}
  & 30''={{\left( \dfrac{30}{60} \right)}^{'}} \\
 & \Rightarrow 30''={{\left( \dfrac{1}{2} \right)}^{'}} \\
\end{align}$
Add the above value in minutes to the minutes we have in the given angle which is $37'$, then the total minutes in the given angle is
$\begin{align}
  & m=37'+{{\left( \dfrac{1}{2} \right)}^{'}} \\
 & \Rightarrow m={{\left( \dfrac{75}{2} \right)}^{'}} \\
\end{align}$
We have the relation $1{}^\circ =60'$. From this we can write the above value in degrees as
$\begin{align}
  & {{\left( \dfrac{75}{2} \right)}^{'}}={{\left( \dfrac{75}{2\times 60} \right)}^{{}^\circ }} \\
 & \Rightarrow {{\left( \dfrac{75}{2} \right)}^{'}}={{\left( \dfrac{5}{8} \right)}^{{}^\circ }} \\
\end{align}$
Now add the above degrees to the degrees we have in the given angle which is $50{}^\circ $, then we will get
$\begin{align}
  & d=50{}^\circ +{{\left( \dfrac{5}{8} \right)}^{{}^\circ }} \\
 & \Rightarrow d={{\left( \dfrac{405}{8} \right)}^{{}^\circ }} \\
\end{align}$
Use the relation $\pi =180{}^\circ $ in the above equation to convert the above degrees into radians, then we will get
$\begin{align}
  & {{\left( \dfrac{405}{8} \right)}^{{}^\circ }}=\dfrac{405}{8}\times \dfrac{\pi }{180} \\
 & \Rightarrow {{\left( \dfrac{405}{8} \right)}^{{}^\circ }}=\dfrac{9\pi }{32} \\
\end{align}$

Hence the value of the angle $50{}^\circ 37'30''$ in radians is $\dfrac{9\pi }{32}$.

Note: In this problem they have mentioned to convert the given angle which is in degrees, minutes and seconds into radians. So, we have reduced the seconds into minutes and minutes into degrees. After that we have converted degrees into radians by using the proper relations. We can also use these relations when we have converted the given radians into degrees.