The math is definitely not fine with turns, because your Euler formula e^ix = cos x + i sin x no longer holds. We can use a base other than
e, namely B = e^2pi which around 535.4916. This doesn't have the nice e properties like d/dx e^x = e^x.
The elegant fact that the base of the natural logarithm, which produces an exponential function that is its own derivative, also shows up as the basis for the above Euler's formula, shows that radians are special: like what binary is to computers.
The natural logarithm being its own derivative is in fact directly linked to the derivative a radians-based sin(x) being cos(x) and so on. Make it any other unit, and you have a mess of conversion factors worse than 2pi.
Imagine complex chained derivatives, double and triple derivative, chain and product rules, all stuffed with trig functions and generating gratuitous piles of cascaded conversion constrants because radians were not used.
The elegant fact that the base of the natural logarithm, which produces an exponential function that is its own derivative, also shows up as the basis for the above Euler's formula, shows that radians are special: like what binary is to computers.
The natural logarithm being its own derivative is in fact directly linked to the derivative a radians-based sin(x) being cos(x) and so on. Make it any other unit, and you have a mess of conversion factors worse than 2pi.
Imagine complex chained derivatives, double and triple derivative, chain and product rules, all stuffed with trig functions and generating gratuitous piles of cascaded conversion constrants because radians were not used.