Question

Express the following angles in radians.
i) ${{30}^{\circ }}$
ii) ${{45}^{\circ }}$
iii) ${{60}^{\circ }}$
iv) ${{90}^{\circ }}$

Answer 1

Hint: We start this question by first considering the formula for converting the angles into radians, ${{1}^{\circ }}=\dfrac{\pi }{180}\ \text{radians}$. Then we consider the given angle in each case and use this formula to find the value of the given angle in radians in each sub part of the question.

Complete step by step answer:
i) ${{30}^{\circ }}$
First let us consider the formula for converting angle from degrees to radians.
As we know that ${{180}^{\circ }}=\pi \ \text{radians}$, then we can write 1 degree as,
${{1}^{\circ }}=\dfrac{\pi }{180}\ \text{radians}$
Now using this formula, we can convert ${{30}^{\circ }}$ into radians as,
\[\begin{align}
  & \Rightarrow {{30}^{\circ }}\times \dfrac{\pi }{{{180}^{\circ }}} \\
 & \Rightarrow \dfrac{\pi }{6} \\
\end{align}\]
So, we get that ${{30}^{\circ }}=\dfrac{\pi }{6}\ \ \text{radians}$
Hence, the answer is \[\dfrac{\pi }{6}\] radians.

ii) ${{45}^{\circ }}$
First let us consider the formula for converting angle from degrees to radians.
As we know that ${{180}^{\circ }}=\pi \ \text{radians}$, then we can write 1 degree as,
${{1}^{\circ }}=\dfrac{\pi }{180}\ \text{radians}$
Now using this formula, we can convert ${{45}^{\circ }}$ into radians as,
\[\begin{align}
  & \Rightarrow {{45}^{\circ }}\times \dfrac{\pi }{{{180}^{\circ }}} \\
 & \Rightarrow \dfrac{\pi }{4} \\
\end{align}\]
So, we get that ${{45}^{\circ }}=\dfrac{\pi }{4}\ \ \text{radians}$
Hence, the answer is \[\dfrac{\pi }{4}\] radians.

iii) ${{60}^{\circ }}$
First let us consider the formula for converting angle from degrees to radians.
As we know that ${{180}^{\circ }}=\pi \ \text{radians}$, then we can write 1 degree as,
${{1}^{\circ }}=\dfrac{\pi }{180}\ \text{radians}$
Now using this formula, we can convert ${{60}^{\circ }}$ into radians as,
\[\begin{align}
  & \Rightarrow {{60}^{\circ }}\times \dfrac{\pi }{{{180}^{\circ }}} \\
 & \Rightarrow \dfrac{\pi }{3} \\
\end{align}\]
So, we get that ${{60}^{\circ }}=\dfrac{\pi }{3}\ \ \text{radians}$
Hence, the answer is \[\dfrac{\pi }{3}\] radians.

iv) ${{90}^{\circ }}$
First let us consider the formula for converting angle from degrees to radians.
As we know that ${{180}^{\circ }}=\pi \ \text{radians}$, then we can write 1 degree as,
${{1}^{\circ }}=\dfrac{\pi }{180}\ \text{radians}$
Now using this formula, we can convert ${{30}^{\circ }}$ into radians as,
\[\begin{align}
  & \Rightarrow {{90}^{\circ }}\times \dfrac{\pi }{{{180}^{\circ }}} \\
 & \Rightarrow \dfrac{\pi }{2} \\
\end{align}\]
So, we get that ${{90}^{\circ }}=\dfrac{\pi }{2}\ \ \text{radians}$
Hence, the answer is \[\dfrac{\pi }{3}\] radians.

Note: The common mistake that happens while solving this question is one might make a mistake of taking the formula for conversion of angle to radians as, ${{1}^{\circ }}=\dfrac{\pi }{360}\ \text{radians}$. But it is wrong because ${{360}^{\circ }}=2\pi $ then we can write it as, ${{1}^{\circ }}=\dfrac{2\pi }{360}=\dfrac{\pi }{180}\ \ \text{radians}$.