Question

Find the degree measure corresponding to the following radian measures $\left( \text{use }\pi =\dfrac{22}{7} \right)$.
${{\left( 1 \right)}^{c}}$

Answer 1

Hint: We will apply here the relation between radians and degrees. This relation is given numerically by ${{\left( \pi \right)}^{c}}={{180}^{\circ }}$. If we divide the expression by $\pi $ to both the denominators the we get the other relation between radians and degree and that is,
$\begin{align}
  & {{\left( \dfrac{\pi }{\pi } \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
 & \Rightarrow {{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
\end{align}$

Complete step-by-step answer:
Now, we will consider the radians ${{\left( 1 \right)}^{c}}$ and we will convert it into its degree. We will do this with the help of the formula which is given by ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$. So, we can directly write ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$ and substitute $\pi =\dfrac{22}{7}$ in this equation to get the answer. This can be done as
$\begin{align}
  & {{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
 & \Rightarrow {{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\dfrac{22}{7}} \right)}^{\circ }} \\
 & \Rightarrow {{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{22}\times 7 \right)}^{\circ }} \\
 & \Rightarrow {{\left( 1 \right)}^{c}}={{\left( \dfrac{90}{11}\times 7 \right)}^{\circ }} \\
 & \Rightarrow {{\left( 1 \right)}^{c}}={{\left( \dfrac{630}{11} \right)}^{\circ }} \\
 & \Rightarrow {{\left( 1 \right)}^{c}}={{\left( 57.\overline{27} \right)}^{\circ }} \\
\end{align}$
The value ${{\left( 57.\overline{27} \right)}^{\circ }}$ is approximately equal to ${{\left( 1 \right)}^{c}}={{57.3}^{\text{o}}}$. Hence, the radian ${{\left( 1 \right)}^{c}}$ is equal to ${{\left( 57.32 \right)}^{\circ }}$ in degrees.

Note: Alternatively we can solve the question as ${{\left( 1 \right)}^{c}}=\left( 1 \right)\times {{\left( 1 \right)}^{c}}$. By substituting the value of ${{\left( 1 \right)}^{c}}$ we will have,
$\begin{align}
  & {{\left( 1 \right)}^{c}}=\left( 1 \right)\times {{\left( 1 \right)}^{c}} \\
 & \Rightarrow \left( 1 \right)\times {{\left( 1 \right)}^{c}}=\left( 1 \right)\times {{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
\end{align}$
This can be written as $\left( 1 \right)\times {{\left( 1 \right)}^{c}}=\left( 1 \right)\times {{\left( \dfrac{180}{\pi } \right)}^{\circ }}$. Therefore we get,
$\begin{align}
  & \left( 1 \right)\times {{\left( 1 \right)}^{c}}=\left( 1 \right)\times {{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
 & \Rightarrow \left( 1 \right)\times {{\left( 1 \right)}^{c}}={{\left( \left( 1 \right)\times \dfrac{180}{\pi } \right)}^{\circ }} \\
 & \Rightarrow \left( 1 \right)\times {{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
\end{align}$
Now we will substitute the value of $\pi =3.14$ approx. Therefore we have,
$\begin{align}
  & \left( 1 \right)\times {{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
 & \Rightarrow \left( 1 \right)\times {{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{3.14} \right)}^{\circ }} \\
 & \Rightarrow \left( 1 \right)\times {{\left( 1 \right)}^{c}}={{\left( 57.32 \right)}^{\circ }} \\
\end{align}$
Hence, the radian ${{\left( 1 \right)}^{c}}$ is equal to ${{\left( 57.32 \right)}^{\circ }}$ in degrees. The radians can also be written directly as substituting 1 radian equal to 57.296 degrees or approximately 1 radian equals to 57.3 degrees. Numerically this can be written as ${{\left( 1 \right)}^{c}}={{57.3}^{\text{o}}}$.