Question

How do you convert $120$ degrees into radians?

Answer 1

In order to convert ${120^ \circ }$ into radians, use the concepts of measurement of angles. From these concepts we know that ${180^ \circ } = \pi$radians. Take out the value of radian for ${1^ \circ }$, then multiply with ${120^ \circ }$, and we get our desired results.

Formula used:
Some formulas used in measurement of angles are:
${180^ \circ } = \pi$radians
$1$radian$= \dfrac{{{{180}^ \circ }}}{\pi }$
$1$ grade $= \dfrac{1}{{100}}$right angles
$\dfrac{D}{{90}} = \dfrac{G}{{100}} = \dfrac{{2R}}{\pi }$

Complete step by step solution:
We are given with ${120^ \circ }$ which is in degrees.
From the concepts of measurement of angles, we know that $180$ in degrees would be equal to $\pi$ radians, that is:
${180^ \circ } = \pi$ radians
Need to find the value of radian for ${1^ \circ }$, for that divide both sides of the upper equation by $180$ and we get:
${180^ \circ } = \pi$ radians
Dividing both the sides by $180$;
$\dfrac{{{{180}^ \circ }}}{{180}} = \dfrac{\pi }{{180}}$radians
${1^ \circ } = \dfrac{\pi }{{180}}$radians
To find the value of ${120^ \circ }$, multiplying the value of $120$ to both sides of the upper equation and we get:
${1^ \circ } \times 120 = 120 \times \dfrac{\pi }{{180}}$radians
${120^ \circ } = \dfrac{{120\pi }}{{180}}$ radians
We can see that the right-hand side fraction can be further simplified.
Since, we know that $120$ and $180$ is both divisible by $60$. $120$ will be cancelled by $60$ and the remainder it gives will be $2$, whereas $180$ will be cancelled by $60$ and the remainder it gives will be $3$, which gives the simplest form. And, from this, the simplest form we get is:
${120^ \circ } = \dfrac{{2\pi }}{3}$radians
Therefore, ${120^ \circ }$ is equal to $\dfrac{{2\pi }}{3}$ radians.

Note:

One radian, written as ${1^c}$, is the measure of the angle subtended at the center of a circle by an arc of length equal to the radius of the circle.
Radian is a constant angle.
The vice versa would occur if the question says to convert radian into degree.