How do I interpret Euler's formula? I don't understand the formula at all:
$$e^{ix} = \cos(x) + i \sin(x)$$
I've tried reading all sorts of webpages and answers on the subject but it's just not clicking with me. I don't understand how we can define things when these are already known quantities. We've already got an exact definition for $e$ as $\lim_{n \to 0} (1+\frac{1}{n})^n$, and we've got $i = \sqrt{-1}$, and we've got $\cos$ and $\sin$ as the $x$ and $y$ coordinates of where right triangles meet the unit circle. 
So I don't understand how we get from those figures to Euler's formula. Then again I also don't understand why complex numbers are of the form $a + bi$ so that might have something to do with it. I have a hard time separating what identities are "emergent" from the previous mathematics and which pieces are "by definition" and why they are defined that way. I don't understand why we start talking about "rotations" as if it's obvious that's what's happening. I don't even know why $e$ is involved in any of this to begin with. 
Is $e^{ix}$ simply being written in the slightly rearranged complex form $a + ib$ where $a = \cos(x)$ and $b = \sin(x)$? Is this to be interpreted as $\cos(x)$ being the $x$-coordinate and somehow $i \sin(x)$ is the $y$ coordinate? 

Ultimately, what we want to do is extend real functions to the complex domain. In principle, this can be done in any arbitrary way, but there is at most only one unique analytic continuation of real functions to the complex plane, see here. As pointed out here, complex differentiable functions are analytic and vice versa, so all we need to do is check if Euler's formula yields a complex differentiable definition of the exponential function.
Any function of the complex variable $z = x + i y$, can be spit up in its imaginary and real parts:
$$f(x + i y) = u(x,y) + i v(x,y)$$
where $u$ and $v$ are real functions of the real variables $x$ and $y$. As explained here, complex differentiability is equivalent to $u$ and $v$ satisfying the Cauchy-Riemann equations. In case of the exponential function, Euler's formula implies:
$\exp(z) = \exp(x)\left[\cos(y) + i \sin(y)\right]\tag{1}$
This obviously reduces to the real exponential function on the real line. It's also easy to check that the pair $u(x,y) = \exp(x)\cos(y)$ and $v(x,y) = \exp(x)\sin(y)$ satisfy the Cauchy-Riemann equations, therefore Eq. (1) is the unique analytic continuation of the real exponential function.
