Question

Find the radian measure corresponding to the following degree measures $\left( \text{use }\pi =\dfrac{22}{7} \right)$.
(i) ${{125}^{\circ }}30'$

Answer 1

Hint: The relation between degrees and radians work better here as we need a conversion for degree to radians. The formula is given numerically by,
$\begin{align}
  & {{\left( \dfrac{\pi }{180} \right)}^{c}}={{\left( \dfrac{180}{180} \right)}^{\circ }} \\
 & \Rightarrow {{\left( \dfrac{\pi }{180} \right)}^{c}}={{\left( 1 \right)}^{\circ }} \\
\end{align}$

Complete step-by-step answer:

(i) We will now consider the degrees ${{125}^{\circ }}30'$ and we will convert it into its simple radians. We know that 1 degree = 60 minutes. This can be numerically written as ${{\left( 1 \right)}^{\circ }}=60'$. Therefore we can write it as ${{\left( \dfrac{1}{60} \right)}^{\circ }}=1'$. Thus we have,
 $\begin{align}
  & {{125}^{\circ }}30'={{125}^{\circ }}+30' \\
 & \Rightarrow {{125}^{\circ }}30'={{125}^{\circ }}+\left( 30\times 1' \right) \\
 & \Rightarrow {{125}^{\circ }}30'={{125}^{\circ }}+\left( 30\times {{\left( \dfrac{1}{60} \right)}^{\circ }} \right) \\
 & \Rightarrow {{125}^{\circ }}30'={{125}^{\circ }}+{{\left( 30\times \dfrac{1}{60} \right)}^{\circ }} \\
 & \Rightarrow {{125}^{\circ }}30'={{125}^{\circ }}+{{\left( \dfrac{1}{2} \right)}^{\circ }} \\
 & \Rightarrow {{125}^{\circ }}30'={{\left( 125+\dfrac{1}{2} \right)}^{\circ }} \\
 & \Rightarrow {{125}^{\circ }}30'={{\left( \dfrac{251}{2} \right)}^{\circ }} \\
\end{align}$
We will solve this with the help of the formula is given by ${{\left( \dfrac{\pi }{180} \right)}^{c}}={{\left( 1 \right)}^{\circ }}$. Therefore, we have ${{\left( \dfrac{251}{2} \right)}^{\circ }}=\dfrac{251}{2}\times {{\left( 1 \right)}^{\circ }}$. By substituting the value of ${{\left( 1 \right)}^{\circ }}$ we will have,
$\begin{align}
  & {{\left( \dfrac{251}{2} \right)}^{\circ }}=\dfrac{251}{2}\times {{\left( 1 \right)}^{\circ }} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}=\dfrac{251}{2}\times {{\left( \dfrac{\pi }{180} \right)}^{c}} \\
\end{align}$
This can be written as ${{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251}{2}\times \dfrac{\pi }{180} \right)}^{c}}$. Therefore we get,
$\begin{align}
  & {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251}{2}\times \dfrac{\pi }{180} \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251\pi }{360} \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251\pi }{360} \right)}^{c}} \\
\end{align}$
Now we will substitute $\pi =\dfrac{22}{7}$ in this equation. Thus, we get
$\begin{align}
  & {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251\pi }{360} \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251}{360}\times \dfrac{22}{7} \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251}{180}\times \dfrac{11}{7} \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{2761}{1260} \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( 2.191 \right)}^{c}} \\
\end{align}$
Hence, we get ${{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{2761}{1260} \right)}^{c}}$ or ${{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( 2.191 \right)}^{c}}$ in decimals.

Hence, the degree ${{125}^{\circ }}30'$ is equal to ${{\left( 2.191 \right)}^{c}}$ in radians.

Note: Alternatively we can put the direct value of ${{\left( 1 \right)}^{\circ }}={{\left( 0.0174 \right)}^{c}}$ in the expression $\begin{align}
  & {{\left( \dfrac{251}{2} \right)}^{\circ }}=\dfrac{251}{2}\times {{\left( 1 \right)}^{\circ }} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}=\dfrac{251}{2}\times {{\left( 0.0174 \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251}{2}\times 0.0174 \right)}^{c}} \\
\end{align}$
At this step we will use BODMAS rule and first solve division operation and after that we will solve multiplication operation. Therefore, we get
$\begin{align}
  & {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( \dfrac{251}{2}\times 0.0174 \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( 125.5\times 0.0174 \right)}^{c}} \\
 & \Rightarrow {{\left( \dfrac{251}{2} \right)}^{\circ }}={{\left( 2.1837 \right)}^{c}} \\
\end{align}$