It's true that this may be one of most startling equations, but its far from being the concoction of the arbitrary approximation in the OP. This is just the spe

It's true that this may be one of most startling equations, but its far from being the concoction of the arbitrary approximation in the OP. This is just the special case of Euler's more general formula e^{ix} = \cos{x} + i\sin{x}, which can easily be derived from Taylor series for e^x, sine, and cosine.