Question

Find the radian measure corresponding to the following degree measures $\left( \text{use }\pi =\dfrac{22}{7} \right)$.
(i) ${{300}^{\circ }}$
(ii) ${{35}^{\circ }}$

Answer 1

Hint: As we know that the relation between radians and degree is always represented by ${{\left( \pi \right)}^{c}}={{180}^{\circ }}$. Therefore by dividing the expression by $180$ to both the denominators the we get the other relation between radians and degree and that is ${{\left( \dfrac{\pi }{180} \right)}^{c}}={{\left( \dfrac{180}{180} \right)}^{\circ }}$ which after simplifying results into ${{\left( \dfrac{\pi }{180} \right)}^{c}}={{\left( 1 \right)}^{\circ }}$.

Complete step-by-step answer:

(i) Now, we will consider the degree ${{300}^{\circ }}$ and we will convert it into radian. We will do this with the help of the formula which is given by ${{\left( \dfrac{\pi }{180} \right)}^{c}}={{\left( 1 \right)}^{\circ }}$. Therefore, we have ${{300}^{\circ }}=300\times {{\left( 1 \right)}^{\circ }}$. By substituting the value of ${{\left( 1 \right)}^{\circ }}$ we will have,
$\begin{align}
  & {{300}^{\circ }}=300\times {{\left( 1 \right)}^{\circ }} \\
 & \Rightarrow {{300}^{\circ }}=300\times {{\left( \dfrac{\pi }{180} \right)}^{c}} \\
\end{align}$
This can be written as ${{300}^{\circ }}={{\left( 300\times \dfrac{\pi }{180} \right)}^{c}}$. Therefore we get,
$\begin{align}
  & {{300}^{\circ }}={{\left( 300\times \dfrac{\pi }{180} \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( 5\times \dfrac{\pi }{3} \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( \dfrac{5\pi }{3} \right)}^{c}} \\
\end{align}$
Now we will substitute $\pi =\dfrac{22}{7}$ in this equation. Thus, we get
$\begin{align}
  & {{300}^{\circ }}={{\left( \dfrac{5\pi }{3} \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( \dfrac{5}{3}\times \pi \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( \dfrac{5}{3}\times \dfrac{22}{7} \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( \dfrac{110}{21} \right)}^{c}} \\
\end{align}$
Hence, we get ${{300}^{\circ }}={{\left( \dfrac{110}{21} \right)}^{c}}$ or ${{300}^{\circ }}={{\left( 5.238 \right)}^{c}}$ in decimals.
(ii) Similarly we will now consider the degree ${{35}^{\circ }}$ and we will convert it into radian. We will do this with the help of the formula which is given by ${{\left( \dfrac{\pi }{180} \right)}^{c}}={{\left( 1 \right)}^{\circ }}$. Therefore, we have ${{35}^{\circ }}=35\times {{\left( 1 \right)}^{\circ }}$. By substituting the value of ${{\left( 1 \right)}^{\circ }}$ we will have,
$\begin{align}
  & {{35}^{\circ }}=35\times {{\left( 1 \right)}^{\circ }} \\
 & \Rightarrow {{35}^{\circ }}=35\times {{\left( \dfrac{\pi }{180} \right)}^{c}} \\
\end{align}$
This can be written as ${{35}^{\circ }}={{\left( 35\times \dfrac{\pi }{180} \right)}^{c}}$. Therefore we get,
$\begin{align}
  & {{35}^{\circ }}={{\left( 35\times \dfrac{\pi }{180} \right)}^{c}} \\
 & \Rightarrow {{35}^{\circ }}={{\left( 7\times \dfrac{\pi }{36} \right)}^{c}} \\
 & \Rightarrow {{35}^{\circ }}={{\left( \dfrac{7\pi }{36} \right)}^{c}} \\
\end{align}$
Now we will substitute $\pi =\dfrac{22}{7}$ in this equation. Thus, we get
$\begin{align}
  & {{35}^{\circ }}={{\left( \dfrac{7\pi }{36} \right)}^{c}} \\
 & \Rightarrow {{35}^{\circ }}={{\left( \dfrac{7}{36}\times \pi \right)}^{c}} \\
 & \Rightarrow {{35}^{\circ }}={{\left( \dfrac{7}{36}\times \dfrac{22}{7} \right)}^{c}} \\
 & \Rightarrow {{35}^{\circ }}={{\left( \dfrac{1}{28}\times \dfrac{11}{1} \right)}^{c}} \\
 & \Rightarrow {{35}^{\circ }}={{\left( \dfrac{11}{28} \right)}^{c}} \\
\end{align}$
Hence, we get ${{35}^{\circ }}={{\left( \dfrac{11}{28} \right)}^{c}}$ or ${{35}^{\circ }}={{0.3928}^{c}}$ in decimals.

Hence, the degree ${{300}^{\circ }}={{\left( \dfrac{110}{21} \right)}^{c}}$ is in radians and the degree ${{35}^{\circ }}={{\left( \dfrac{11}{28} \right)}^{c}}$ is in radians.

Note: Alternatively we can solve it directly substituting $\left( \pi \right)$ as $3.14$. By this we will get the solution in the way as done below.
$\begin{align}
  & {{300}^{\circ }}={{\left( \dfrac{5\pi }{3} \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( \dfrac{5}{3}\times 3.14 \right)}^{c}} \\
\end{align}$
For solving it further we will use BODMASS rule in which we will divide first and then multiply the terms together. Thus we get
$\begin{align}
  & \Rightarrow {{300}^{\circ }}={{\left( 1.66\times 3.14 \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( 5.2124 \right)}^{c}} \\
\end{align}$
Alternatively we can put the direct value of ${{\left( 1 \right)}^{\circ }}={{\left( 0.0174 \right)}^{c}}$ in the expression $\begin{align}
  & {{300}^{\circ }}=\left( 300 \right)\times {{\left( 1 \right)}^{\circ }} \\
 & \Rightarrow {{300}^{\circ }}=\left( 300 \right)\times {{\left( 1 \right)}^{\circ }} \\
 & \Rightarrow {{300}^{\circ }}=300\times {{\left( 0.0174 \right)}^{c}} \\
 & \Rightarrow {{300}^{\circ }}={{\left( 5.22 \right)}^{c}} \\
\end{align}$
Similarly we can apply this procedure to ${{35}^{\circ }}={{\left( \dfrac{11}{28} \right)}^{c}}$.