Question

How do you convert 5 radians to degrees?

Answer 1

Hint: We are used to measuring angles in degrees. The whole notion of degrees is a human constructed system. It is not the only way in which you can measure angles. A full rotation or a complete angle is measured as ${{360}^{\circ }}$. Hence, we shall look at the basic conversions to find the relation between degree and radians.

Complete answer:
One of the reasons behind it are ancient calendars. They were based on 360 days in a year. Some ancient astronomers observed that things seemed to move $\dfrac{1}{360}$ of the sky per day. Another theory is that the ancient Babylonians liked equilateral triangles a lot and they had a base 60 number system. So, they had 60 symbols. But we have only 10 symbols. We have a base 10 number system.
The system of radians of measuring the angles is much more mathematically pure than degrees. Its not based on these cultural artifacts of base 60 number systems or astronomical patterns.
$2\pi $ radians $=360$degrees for one full revolution.
Now, dividing both sides by 2, we get
$\pi $ radians $=180$degrees
We shall now divide both sides by $\pi $,
$\Rightarrow 1$ radians $=\dfrac{180}{\pi }$ degrees
$\Rightarrow 5$ radians $=5.\dfrac{180}{\pi }$ degrees
$\therefore 5$ radians $=\dfrac{900}{\pi }$ degrees

Note: The system of measuring angle in radians is more accurate and useful because it tells you exactly the arc length that is subtending the angles. The arc that subtends the angle of full rotation is the entire circumference of the circle thus formed. The angle is called $2\pi $ radians as it is subtended by an arc length of $2\pi $ radii.