Question on $\cos(x)\in\mathbb{C}$ If we have $-1\le\cos(x)\le1$ and $\cos(ix)=\frac{e^x+e^{-x}}2$, then both are defined as real for real $x$.
This leads me to the conclusion that $\cos(x)\in\mathbb{R}$ if $x$ is real or imaginary.
However, I have to imagine that $\cos(x)\in\mathbb{C}$ is possible, but how? 

$x$ doesn't have to be real or imaginary, for example $\cos(1+i)$ is not real or imaginary.
