Question

Find the degree measure corresponding to the following radian measures $\left( \text{use }\pi =\dfrac{22}{7} \right)$.
(ii) ${{\left( \dfrac{8\pi }{5} \right)}^{c}}$

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:

We will consider the radians ${{\left( \dfrac{8\pi }{5} \right)}^{c}}$ and we will convert it into its degree. We will do this with the help of the formula which is given by ${{\left( 1 \right)}^{c}}={{\left( \dfrac{180}{\pi } \right)}^{\circ }}$. Therefore, we have ${{\left( \dfrac{8\pi }{5} \right)}^{c}}=\dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}$. By substituting the value of ${{\left( 1 \right)}^{c}}$ we will have,
$\begin{align}
  & {{\left( \dfrac{8\pi }{5} \right)}^{c}}=\dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}} \\
 & \Rightarrow \dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}=\dfrac{8\pi }{5}\times {{\left( \dfrac{180}{\pi } \right)}^{\circ }} \\
\end{align}$
This can be written as $\dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}={{\left( \dfrac{8\pi }{5}\times \dfrac{180}{\pi } \right)}^{\circ }}$. By cancelling the $\pi $ both from numerator and denominator we will have,
$\begin{align}
  & \dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}={{\left( \dfrac{8\pi }{5}\times \dfrac{180}{\pi } \right)}^{\circ }} \\
 & \Rightarrow \dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}={{\left( \dfrac{8}{5}\times \dfrac{180}{1} \right)}^{\circ }} \\
 & \Rightarrow \dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}={{\left( \dfrac{8}{1}\times \dfrac{36}{1} \right)}^{\circ }} \\
 & \Rightarrow \dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}={{\left( 8\times 36 \right)}^{\circ }} \\
 & \Rightarrow \dfrac{8\pi }{5}\times {{\left( 1 \right)}^{c}}={{\left( 288 \right)}^{\circ }} \\
\end{align}$
 Hence, the radian ${{\left( \dfrac{8\pi }{5} \right)}^{c}}$ is equal to ${{\left( 288 \right)}^{\circ }}$ in degrees.

Note: Alternatively we can directly get the value by substituting 1 radian equal to 57.296 degrees or approximately 1 radian equals 57.3 degrees. Numerically this can be written as ${{\left( 1 \right)}^{c}}={{57.3}^{\text{o}}}$. A radius of any circle moves from one point to the other inside the circle only then it forms an arc. And because of that we get radians and degrees of a circle. The difference between them is that 57.3 degrees together makes one radians. And 0.0175 radians form 1 degree. The relationship between them is given numerically as ${{\left( 1 \right)}^{c}}={{57.3}^{\text{o}}}$ and ${{\left( 1 \right)}^{\text{o}}}={{0.0175}^{c}}$. Apart from degrees and radians we also have a term known as gradient and the relationship between gradient and degree is given by ${{90}^{\text{o}}}={{100}^{g}}$ which means that 100 gradients together form 90 degrees angle.