Lucine M. answered • 03/23/19

UCLA Grad & Experienced Math Teacher/Tutor (15+ years)

Hi there! To answer your question, first and foremost, when integrating or differentiating trig functions, we don't *have to* use radians, but rather, we choose to. There's nothing wrong with nor impossible about using degrees as the way to measure angles. However, mathematicians use radians because it is a more elegant and natural way to measure angles than degrees. Have you ever wondered why a circle is 360^{o}? The reason is because it was decided by a civilization that used a base-6 number system (we use a base-10 number system). The choice to have a circle be 360^{o} is completely arbitrary. Radians on the other hand, are based on the idea of a radius angle: an angle created on a circle by using an arc length equal to the length of the radius of the circle. Why is a circle 2π radians? Because we can fit 2π radii into the length of the circumference (C = 2 π r). I hope that helps!