Radians have the property that if we step x by some tiny amount δ, then the cos/sin coordinates will move by that same distance around the unit circle:

   |[cos(x+δ) + i sin(x+δ)] - [cos(x) + i sin(x)]| = δ

This is also related to how we can estimate sin(x) = x for small values next to zero, if using radians.

In radians, the derivative sin'(x) is cos(x), and cos'(x) is -sin(x). Derviation just shifts the waveform left by ninety degrees. In units other than radians, we get wacky constant terms that change at each step.

That's related to how e^x is its own derivative.