Advantages of having radian measure

Answer 1

There are many; for example,

sin(x) = x - x^3/3! + x^5/5^ - x^7/7! ...

cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ...

are valid only when angles are measured in radians.

The derivatives of trigonometric functions

d/dx sin(x) = cos(x),

d/dx cos(x) = -sin(x) as well as the derivatives of other trigonometric ratios are valid (in relatively simple forms) only if defined in radians. The same applies to integrals of these functions.