Why are trig-functions standard in radians as opposed to degrees?
I was wondering about this because when differentiating a trig-function in degrees one should apply the chain-rule: $$\frac d{dx}\sin x^\circ=\frac\pi{180}\cos x^\circ$$ Why could the degrees not be the standard?
Degrees have no mathematical basis, actually it is not known with certainty where the measure originates from (though apparently it may be related to the fact that a year has approx. 360 days, meaning the earth rotates around the sun by ~1deg each day).
Radians however, make perfect mathematical sense. They represent the length of the arc delimited by the corresponding angle on a radius 1 circle.
It also turns out this makes most computations much more easy, in particular with complex numbers : the cosine and sine functions (and other trig functions) have simpler expressions in terms of the exponential function. Their Taylor expansions don't have pi's all over the place. Their derivatives are easy to express as a function of each other ; and the list goes on.