Question

How do you express $\dfrac{{11\pi }}{{12}}$ in degrees?

Answer 1

Hint: The above angle is given in radians and has to be converted into degrees. To express radians in degrees, you have to first find the conversional formula to convert radians in degrees, and then convert the radians into degrees with help of the derived formula. To find the relation between them, remember the whole angle of a circle, we know that a circle has $2\pi $ angle in radians and ${360^0}$ angle in degrees. Compare these two to get the required conversional formula of radians into degrees.

Complete step by step answer:
To convert $\dfrac{{11\pi }}{{12}}$ radians to degrees, we need the conversional formula by which we will convert radians in degrees. We should take the help of the angle of a circle to find the relation between degrees and radians. In a circle, we know that the complete angle is equal to $2\pi $ angle in radians and ${360^0}$ angle in degrees. So we can write it mathematically as follows
\[2\pi \;{\text{radians}} = 360\;{\text{degrees}} \\
\Rightarrow 1\;{\text{radian}} = \dfrac{{360}}{{2\pi }}{\text{degrees}} \\
\Rightarrow 1\;{\text{radian}} = \dfrac{{180}}{\pi }{\text{degrees}} \\
\Rightarrow x\;{\text{radians}} = \dfrac{{180x}}{\pi }{\text{degrees}} \\ \]
Now we have the conversion formula for converting radians into degrees, that is
\[ x\;{\text{radians}} = \dfrac{{180x}}{\pi }{\text{degrees}}\]
So converting $\dfrac{{11\pi }}{{12}}$ radians to degrees with the help of the derived formula, we get
$x\;{\text{radians}} = \dfrac{{180x}}{\pi }{\text{degrees}} \\
\Rightarrow \;\dfrac{{11\pi }}{{12}}\;{\text{radians}} = \dfrac{{180}}{\pi } \times \dfrac{{11\pi }}{{12}}{\text{degrees}} \\ $
Simplifying it further,
$\Rightarrow \;\dfrac{{11\pi }}{{12}}\;{\text{radians}} = \dfrac{{180}}{\pi } \times \dfrac{{11\pi }}{{12}}{\text{degrees}} \\
\therefore \;\dfrac{{11\pi }}{{12}}\;{\text{radians}} = 165\;{\text{degrees}}\;{\text{or}}\;16{{\text{5}}^0} \\ $
Therefore $\dfrac{{11\pi }}{{12}}$ radians equals ${165^0}$.

Note:To convert degrees into radians derive the conversion formula by the same procedure with the help of the whole angle of the circle. Derive this conversional formula and check your answer with below: $x\;{\text{degrees}} = \dfrac{{x\pi }}{{180}}\;{\text{radians}}$
 Degrees and radians are commonly used units for the measurement of angles. Degrees can be further divided into minutes and seconds whereas radian is the S.I. (Standard International) unit for measurement of the angles.