[cka]

Further evidence that e^{ix} = cos(x) + i sin(x) is natural is that it fits in nicely with power series representations. If you define cos(x), sin(x), and e^x by their power series centered at 0 then it's straightforward to see that substituting ix into the power series for e^x yields the sum of the power series for cos(x) and i*sin(x) (as long as you accept that theorems about absolute convergence and rearranging terms extend to the complex numbers).

[tonyarkles]

The lecture in my Calculus IV course where we walked through that derivation is literally my only memorable math lecture in university. It just so beautifully ties so many different concepts together!