How do complex number exponents actually work? I know Euler's formula and how to take complex exponents, but in it it's $e$ to an imaginary angle, not a number, it seems. From my understanding pi itself is not an angle, but $\pi$ radians is. And since cosine can only take in an angle, or at least a representation of one, and the variable is the same everywhere in Euler's formula, the exponent should be an angle and not a number. The question is then raised could I then say $e^{180i}=-1$? Surely this has a single numerical value though, right? 
I've been bothered by this for a long time. Am I wrong or how can I take an exponent with a complex number? 
Edit for the duplicate: I understand how to take them and how they're supposed work and be justifyed/proved, but raising something to the power of an angle which is what I thought was meant to be happening made me question how they actually worked that could allow this. 

A number of people are giving fairly complex answers (no pun intended) about exponentiation, but I want to address a much simpler misunderstanding in your answer, which should hopefully clear up some of the issues.

And since cosine can only take in an angle

No, the trig functions definitely take numbers. They're related to the circle, in that they're cyclic, and the "width" of their repetition is exactly  2π, which happens to be the circumference of a circle when you measure it in radians. But the argument to trig functions aren't angles except by convention.

The question is then raised could I then say e180i=−1?

180 isn't equal to π, any more than it's equal to 50 just because 180 degrees is 50% of circle. 180 degrees is equal to π radians, but since the exponentiation operator also takes numbers, not angles, that equivalence between angle units isn't relevant.
