Question

How do you convert $80$ degrees into radians?

Answer 1

Hint:First find the relation between degrees and radians, and then convert the given degrees into the radian form. In order to find the relation between degrees and radians, refer to the whole angle of a circle, we know that a circle has ${360^0}$ angle in degrees and $2\pi $ angle in radians. Compare these two and get the required relation for conversion of the degrees into the radians.

Complete step by step solution:
In order to convert $80$ degrees into radians, we need the
conversional formula through which we can change degrees into radians. We should take the help of a circle in order to find the relation between degrees and radians.
In a circle the complete angle is equal to ${360^0}$ angle in degrees and $2\pi $ angle in radians. So we can write it mathematically
$
\Rightarrow 360\;{\text{degrees}} = 2\pi \;{\text{radians}} \\
\therefore 1\;{\text{degree}} = \dfrac{{2\pi }}{{360}}{\text{radians}} \\
\Rightarrow 1\;{\text{degree}} = \dfrac{\pi }{{180}}\;{\text{radians}} \\
\therefore x\;{\text{degrees}} = \dfrac{{x\pi }}{{180}}\;{\text{radians}} \\
$
So we have got the conversion formula for converting degrees into radians, that is $x\;{\text{degrees}}
= \dfrac{{x\pi }}{{180}}\;{\text{radians}}$
so converting $80$ degrees into radians with the help of the derived formula,
$
\Rightarrow x\;{\text{degrees}} = \dfrac{{x\pi }}{{180}}\;{\text{radians}} \\
\Rightarrow 80\;{\text{degrees}} = \dfrac{{80\pi }}{{180}}\;{\text{radians}} \\
$
Simplifying it further,
$
\Rightarrow 80\;{\text{degrees}} = \dfrac{{80\pi }}{{180}}\;{\text{radians}} \\
\Rightarrow 80\;{\text{degrees}} = \dfrac{{4\pi }}{9}\;{\text{radians}} \\
$
Therefore $80$ degrees equals to $\dfrac{{4\pi }}{9}\;{\text{radians}}$

Note: Degrees and radians are commonly used unit for measurement of angles in degrees unit system a circle is divided into $360$ equal parts, that’s why a right angle or a quarter circle has angle of $\dfrac{{360}}{4} = 90$ degrees. Degrees are also further divided into minutes and seconds.
Radians are the derived S.I. (Standard International) unit of angles, a radian is defined as the angle for which the arc length is equal to the length of radius. The circumference of a unit circle is $ = 2\pi $ and so the total angle of the circle is $ = 2\pi $ in radians.