Question

How do you express ${{220}^{\circ }}$ in radian measure?

Answer 1

Hint: For converting the angle given in the above question, which is ${{220}^{\circ }}$, given in the degree measure, we have to use the relation between the degree and the radian measures. The radian is related to the degree by the relation $\pi \text{ rad}={{180}^{\circ }}$. Using this relation we can determine the radian angle equivalent to one degree as ${{1}^{\circ }}=\dfrac{\pi }{180}\text{rad}$. Using the unitary method, we can finally determine the radian angle equivalent to the given angle in degrees, which is equal to ${{220}^{\circ }}$.

Complete step-by-step solution:
We know that the measure of radian is related to the measure of degree by the relation
$\Rightarrow \pi \text{ rad}={{180}^{\circ }}$
Or
$\Rightarrow {{180}^{\circ }}=\pi \text{ rad}$
Dividing both the sides by $180$, we get
\[\begin{align}
  & \Rightarrow \dfrac{{{180}^{\circ }}}{180}=\dfrac{\pi }{180}\text{ rad} \\
 & \Rightarrow {{1}^{\circ }}=\dfrac{\pi }{180}\text{ rad} \\
\end{align}\]
The angle given in the above question is equal to ${{220}^{\circ }}$. Therefore, using the unitary method, we multiply both the sides of the above equation by $220$ to convert the given angle in radians. So the above equation becomes
$\begin{align}
  & \Rightarrow 220\times {{1}^{\circ }}=220\times \dfrac{\pi }{180}\text{rad} \\
 & \Rightarrow {{220}^{\circ }}=\dfrac{220\pi }{180}\text{rad} \\
\end{align}$
Cancelling the numerator and the denominator by $20$ on the right hand side of the above equation, we get
$\Rightarrow {{220}^{\circ }}=\dfrac{11\pi }{9}\text{rad}$
Hence, the given angle of ${{220}^{\circ }}$ is expressed in the radian measure as $\dfrac{11\pi }{9}\text{rad}$.

Note: For solving this question, the relation between the degree and the radian measurements given by $\pi \text{ rad}={{180}^{\circ }}$ must be remembered very much carefully. We must remember that pie radians are equated to the degrees in this relation, and not one radian. Also, we must not reverse the numbers $\pi $ and $180$ in this relation out of confusion.