Shouldn't I be able to use both rads and degrees in complex exponentials?

Question

Up until now, I've been using rads and degrees interchangeably, simply using the $^{\circ}$ symbol to signify degrees, and then using the correct trigonometric function, so that: $$sin(90^\circ)=sin(π/2)$$

I would think that the same line of thought could be used when dealing with complex exponentials, since x appears to always end up in a trigonometric function: $$e^{ix}=cos(x) + isin(x)$$

However this seems to completely break down when logarithms are brought into the picture: $$ln(e^{ix})=i(x + 2κπ),\hspace{1em}κ\in\mathbb{Z}$$

But (assume $κ=0$ for simplicity's shake) $$ln(5e^{i90^\circ}) = ln5 + i90^\circ$$ Isn't the same number as $$ln(5e^{iπ/2}) = ln5 + iπ/2$$

My textbook (on electronic circuit analysis) tells me to use radians here, but there is no mention as to why. Any help would be greatly appreciated.

Answer 1

Why it matters:

In degrees, Euler's formula would read

$$e^{iz\pi/180}=\cos(z)+i\sin(z)$$ and the whole world would hate that $\frac\pi{180}$ factor.

Answer 2

The degree is a very arbitrary unit. It is basically just 1/360 of a circle. I believe this unit was developed to be based on the rotation of the Earth around the sun. Past measurement techniques approximated the orbit to be about 360 days.

The radian is a much more specific measurement that tends to be easier for conversion and calculation. One radian is defined as the angle of a circle subtended by the radius and an arc length equal to the radius. So for example, in a circle of radius 1 meter, the angle subtended by an arc of length 2 meters is 2/1 = 2 radians.

The questions mathematicians usually answer correlate with the radian instead of the degree.