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Question: How do you convert \(80\) degrees into radians?...

How do you convert 8080 degrees into radians?

Explanation

Solution

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 3600{360^0} angle in degrees and 2π2\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 8080 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 3600{360^0} angle in degrees and 2π2\pi angle in radians. So we can write it mathematically
⇒360  degrees=2π  radians ∴1  degree=2π360radians ⇒1  degree=π180  radians ∴x  degrees=xπ180  radians  \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  degrees=xπ180  radiansx\;{\text{degrees}} = \dfrac{{x\pi }}{{180}}\;{\text{radians}}
so converting 8080 degrees into radians with the help of the derived formula,
⇒x  degrees=xπ180  radians ⇒80  degrees=80π180  radians  \Rightarrow x\;{\text{degrees}} = \dfrac{{x\pi }}{{180}}\;{\text{radians}} \\\ \Rightarrow 80\;{\text{degrees}} = \dfrac{{80\pi }}{{180}}\;{\text{radians}} \\\
Simplifying it further,
⇒80  degrees=80π180  radians ⇒80  degrees=4π9  radians  \Rightarrow 80\;{\text{degrees}} = \dfrac{{80\pi }}{{180}}\;{\text{radians}} \\\ \Rightarrow 80\;{\text{degrees}} = \dfrac{{4\pi }}{9}\;{\text{radians}} \\\
Therefore 8080 degrees equals to 4π9  radians\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 360360 equal parts, that’s why a right angle or a quarter circle has angle of 3604=90\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 =2π= 2\pi and so the total angle of the circle is =2π= 2\pi in radians.