Why do we require radians in calculus?

Question

I think this is just something I've grown used to but can't remember any proof. When differentiating and integrating with trigonometric functions, we require angles to be taken in radians. Why does it work then and only then?

Answer 1

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!

Answer 2

There is no such requirement....

Regardless of the mathematical operation involved, you can always use either radians or degrees. You simply have to define what you are using, and write the equations accordingly.

Suppose I want the rms value of a sine wave: rms (root mean square) is defined as the square root of (the integral of sin^2 of x, divided by the period)

If you define x as being in radians, then the definite integral is taken from 0 to 2*pi. If x is in degrees, then 0 to 360

Answer 3

Angle measure is arbitrary. Why is a right angle 90° and not 127?

The use of radians allows a direct relation between angular measure and linear measure.