When I was a physics TA, I used similar arguments with double derivatives show the connection between imaginary exponentials and sines/cosines.

For f(x)=e^(kx), double derivative d^2f/dx^2 = k^2 f(x)

Meanwhile for g(x)=sin(kx), d^2g/dx^2 = -k^2 g(x), and similarly for cosine.

So if k is imaginary, from a differential equations point of view, the exponential behaves exactly like a sine or cosine.

That shows the general idea, and further consideration of boundary conditions gives e^ix = cos(x) + isin(x).