Why use radians when differentiating trig functions?

Answer 1

When it comes to differentiating, polar coordinates are more convenient because a derivative gives the slope of the tangent to a curve at some specific point. Since a curve is round, it makes more sense to use a circular approximation than a rectangular approximation, for sake of accuracy and precision. This doesn't make too much of a difference in terms of derivatives, but once you start hitting integrals, differentials, partials derivatives, and gradients of multivariable functions, you'll definitely want to be using radians instead of degrees.

Also, radians are generally preferred because they are all related by a factor of pi, a natural number, instead of 180 degrees, an arbitrary number, which makes them easier to calculate without a calculator (than degrees).