Ok but who wants to use e^ix\180/pi) = cos(x*180/pi) + i*sin(x*180/pi) when you could just use radians and use e^iz = cos(z) + i sin(z), which is valid for all complex numbers?

Also, the derivative of sin(x*180/pi) satisfies the differential equation

x'' = -(180/pi)^2 x

which is ugly and arbitrary. If use radians, you have that sin(x) satisfies

x'' = -x

which is neat and intuitive.

Even worse, the taylor series for sin(x*180/pi) is ugly and has tons of unnecessary constants, and doesn't naturally pop up anywhere in math like sin(x) does.

If you want to use degrees in mathematics, you end up with a mess of ugly terms and constants. Measurements in degrees are artificial and arbitrary. Measurements in radians are natural and mathematically elegant. The number pi is ubiquitous in mathematics, but degrees are not.