Question

What is $\dfrac{\pi }{8}$ in degrees?

Answer 1

Hint: In the above question, we have given $\dfrac{\pi }{8}$ and are asked to convert it into degrees. For that we are going to write $\pi $ in degrees and we know that $\pi ={{180}^{\circ }}$ so substituting the value of $\pi $ in $\dfrac{\pi }{8}$ and then simplify the fraction by dividing the numerator and the denominator by the common factors.

Complete step by step solution:
In the above problem, we are asked to convert $\dfrac{\pi }{8}$ in degrees. As you can see that $\dfrac{\pi }{8}$ is in the radian form because $\pi $ is given. Now, to convert this $\dfrac{\pi }{8}$ in degrees we are going to substitute $\pi $ in degrees.
We know that $\pi ={{180}^{\circ }}$ so substituting this value of $\pi $ in $\dfrac{\pi }{8}$ we get,
$\dfrac{{{180}^{\circ }}}{8}$
Now, we are going to simplify the above fraction. As you can see that the numerator and the denominator are divisible by 2 so dividing ${{180}^{\circ }}$ by 2 and 8 by 2 we get,
$\dfrac{{{90}^{\circ }}}{4}$
Now, again you can see that ${{90}^{\circ }}$ is divided by 2 and 4 is also divided by 2 so dividing the numerator and denominator by 2 we get,
$\dfrac{{{45}^{\circ }}}{2}$
Now, solving the above fraction we are going to divide ${{45}^{\circ }}$ by 2 using long division method we get,
$2\overset{22.5}{\overline{\left){\begin{align}
  & 45 \\
 & 4 \\
 & \overline{0}5 \\
 & 04 \\
 & \overline{\begin{align}
  & 010 \\
 & 010 \\
 & \overline{000} \\
\end{align}} \\
\end{align}}\right.}}$
From the above division, the result of $\dfrac{{{45}^{\circ }}}{2}$ is equal to ${{22.5}^{\circ }}$.

Hence, we have converted $\dfrac{\pi }{8}$ into ${{22.5}^{\circ }}$.

Note: In the above problem, we have converted radian into degrees. Similarly, we could have asked to convert the degree into radians. Lets say, we are asked to convert ${{120}^{\circ }}$ into radians.
We know that ${{180}^{\circ }}=\pi $ so by unitary method we can find the value of ${{1}^{\circ }}$ in radians by dividing $\pi $ by ${{180}^{\circ }}$ then we get,
${{1}^{\circ }}=\dfrac{\pi }{180}$
So, for ${{120}^{\circ }}$ multiplying 120 on both the sides of the above equation and we get,
$\begin{align}
  & {{1}^{\circ }}\times 120=\dfrac{\pi }{180}\times 120 \\
 & \Rightarrow {{120}^{\circ }}=\dfrac{\pi }{180}\times 120 \\
 & \Rightarrow {{120}^{\circ }}=\dfrac{2\pi }{3} \\
\end{align}$
Hence, we have converted ${{120}^{\circ }}$ into $\dfrac{2\pi }{3}$.