Question

How do you express $132$ degrees in radians?

Answer 1

We are given with an angle in degrees which we have to express in radians. We know that, ${{180}^{\circ }}=\pi \text{ radians}$ as $2\pi$ would mean a complete circle which is ${{360}^{\circ }}$. So, ${{1}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}\text{ radians}$, we will multiply the given angle in degrees by $\dfrac{\pi }{{{180}^{\circ }}}$ to express the angle in radians. Reducing it further, cancelling the common terms, we will have the angle in radians.

Complete step-by-step solution:
According to the given question, we have been given an angle which is expressed in degrees, that is, we have $132$ degrees. We have to now express this angle in radians.
We will begin with writing the given angle in degrees, we have,
$132$ degrees -----(1)
We know that a complete circle measures ${{360}^{\circ }}$ which in radian terms would be $2\pi$. So, for a half circle that is ${{180}^{\circ }}$ we have the angle in radian terms as $\pi \text{ radians}$.
We can now write it as:
${{360}^{\circ }}=2\pi \text{ radians}$
${{180}^{\circ }}=\pi \text{ radians}$
So, for 1 degree we have,
${{1}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}\text{ radians}$
We will use this conversion factor to get the angle in the required unit, that is, we will multiply the given angle in degrees by $\dfrac{\pi }{{{180}^{\circ }}}$ and further solving which gives us the angle in radians.
We have,
If ${{1}^{\circ }}\to \dfrac{\pi }{{{180}^{\circ }}}\text{ radians}$
Then for $132$ degrees,
$\Rightarrow {{132}^{\circ }}\to \dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\text{ radians}$
We will now solve $\dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\text{ radians}$ and we get,
$\Rightarrow \dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\text{ radians}$
We see that, numerator and the denominator have a multiple of 12. So, we will divide and multiply by 12 and reduce the expression, we have,
$\Rightarrow \dfrac{\pi }{{{180}^{\circ }}}\times {{132}^{\circ }}\times \dfrac{12}{12}$
Reducing the terms, we get,
$\Rightarrow \dfrac{\pi }{15}\times 11$
$\Rightarrow \dfrac{11\pi }{15}$
Therefore, we get, ${{132}^{\circ }}=\dfrac{11\pi }{15}\text{ radians}$.

Note: The conversion factor for degrees to radians is ${{1}^{\circ }}=\dfrac{\pi }{{{180}^{\circ }}}\text{ radians}$.
Similarly, the conversion factor for radians to degrees is $1 \text{ radians}=\dfrac{{{180}^{\circ }}}{\pi }\deg .$
The conversion factor should be carefully written and calculation should be done in a proper sequence to avoid errors.