Why $e^{iπ}=-1$ and not $e^{180i}=-1$?

Question

I know that $e^{iπ}=-1$ is the outcome of the identity $e^{ix}=cos(x)+isin(x)$ when $x=π$. What I don't understand is: why radians? Wouldn't it also be true that $e^{180i}=-1$ - just in different units? And wouldn't the identity $e^{i\pi}=-1$, then, only be true when $-1$ is the number of radians?

Answer 1

These two are in different units! The measure of an angle is either expressed in radians or degrees. The notations used is $2\pi$ radians is one full round (circle), which is same as $360^\circ$.

Thus $\pi$ radians is exactly same as $180^\circ$.