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. 

Just a little note that I hope can be of some use. In order to avoid confusion when dealing with angles and unit of measures of them, one can directly define the trigonometric functions starting with the arc lenght measurement.
Consider the semicircle of equation 
$$f(x) = \sqrt{1-x^2}, \ \ x\in [-1,1].$$
It is relatively easy to show that the arc length between point $(y, f(y))$ and point $(1,0)$ can be computed with the (improper) integral
$$A(y) = \int_y^1\frac{1}{\sqrt{1-x^2}}dx, \ \ y\in [-1, 1].$$
Here the unit of measure is clearly the same as the one used for determining abscissae and ordinates.
$A(y)$ is continuous in $[-1,1]$, differentiable in $(-1,1)$, and strictly decreasing. So it has a well defined inverse function
$$A^{-1}(x)= \cos(x)$$
with domain $[0, \pi]$, where, by definition $A(-1) = \pi$. The rest of the cosine function can be defined using appropriate shifts and periodicity.
