Question

What is the angle $\left( \dfrac{\pi }{10} \right)$ in degrees?

Answer 1

We need to know the relation between radians and degrees. The relation between the degrees and radians is $180{}^\circ =\pi \text{ radians}\text{.}$ Therefore, 1 radian is given by $\text{1 radians}=\dfrac{180{}^\circ }{\pi }\text{.}$ Using this, we calculate the value of $\left( \dfrac{\pi }{10} \right)$ in degrees.

Complete step-by-step solution:
To solve this question, we consider the relation between degrees and radians as,
$\Rightarrow 180{}^\circ =\pi \text{ radians}$
This is obtained from the fact that the angle subtended by an arc in one full rotation around the circle is $360{}^\circ .$ Representing the same using radians, it is given by $2\pi \text{ radians}\text{.}$
Dividing both sides of the first equation by $\pi ,$ we can find the value of 1 radian. 1 radian can be represented by,
$\Rightarrow \text{1 radians}=\dfrac{180{}^\circ }{\pi }$
The given question has the value $\left( \dfrac{\pi }{10} \right)$ radians. Multiplying this term with both sides of the above equation,
$\Rightarrow \left( \dfrac{\pi }{10} \right)\text{ radians=}\left( \dfrac{180{}^\circ }{\pi } \right)\times \left( \dfrac{\pi }{10} \right)$
Calculate the right-hand side of the equation. This can be done by cancelling the common factors in the numerator and denominator. The $\pi$ terms can be cancelled and 180 divided by 10 is 18.
$\Rightarrow \left( \dfrac{\pi }{10} \right)\text{ radians=}18{}^\circ$
This means that for a rotation of $\dfrac{\pi }{10}\text{ radians,}$ it is equivalent to moving $18{}^\circ$ in the circle.
Hence, the value of $\left( \dfrac{\pi }{10} \right)$ in degrees is $18{}^\circ .$

Note: It is important to know the concepts of radians and degrees and their relations and interconversions. These two are nothing but the units in which the angle subtended by an arc of a circle is measured in. One full circle has an angle of $360{}^\circ .$ This means that an arc rotating inside the circle covers an angle of $360{}^\circ .$ We can also represent this using radian which is given by $2\pi \text{ radians}\text{.}$ This concept forms the basis for many trigonometrical problems.