$e^{iθ}$

Once one sees the series expansions of $\cos θ$, $\sin θ$, and $e^x$, it's easy to see why
$e^{iθ} = \cos θ + i\sin θ$,
and from there, why
$e^{iπ} + 1 = 0$
which I always thought appeared mystical.

However, I'm not satisfied; I'd like to understand it at a visual level. Also, I'd like to understand how to derive the related Taylor/Maclaurin series.

I learned Gaussian integers ($m + ni, | m, n ∈ ℤ$) and proved an equilateral triangle cannot have all vertices at Gaussian integers.