complex numbers - Why is $icdot sin(x)$ not $cos(x)$?

I recently repeated some math basics of the Fourier transform and of course stumbled across Euler's formula. When reading the term $\cos(x) + i\sin(x)$ I wondered why it could not be written as $2\cos(x)$.

Since all professors always emphasize that a cosine is nothing but a $90$ degree shifted sine, I was wondering why the multiplication with i, which also causes a $90°$ shift on the complex plane, doesn't result in a $\cos$-function.