[kazinator]

If we use a degrees version of sin and cos (call them sind and cosd), then we cannot have e on the left side without a conversion factor.

      (iπx/180)
     e            = cosd x + i sind x

            ix
      π/180 
  -> e            = cosd x + i sind x


                   π/180 
  -> let   f =   e

       ix
     f            = cosd x + i sind x

Probem is, f doesn't have nice properties like:

  d    x             x
  -  f        /=   f
  dx

There is something uniquely special about the unit circle, and about using the unscaled distance around the unit circle as the measure of the angle.

[kazinator]

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.