Say there's a sinusoid wave with amplitude $A$, frequency $\omega$, and phase shift $\psi$, then one way to write it is $A cos(\omega t - \psi)$. But it can also be written as $Re(Ae^{i(\omega t - \psi)})$, and using this complex-exponential form seems to be common in engineering, physics, and anywhere with fourier/laplace transforms.
I understand why the real part of the form above is a cosine from Euler's formula $e^{i\theta} = cos(\theta) + isin(\theta)$, and I understand the geometric intuition of complex numbers as points on a plane.
However, what information are we losing by throwing away the imaginary part of the complex-exponential? $Im(Ae^{i(\omega t - \psi)}) = Asin(\omega t - \psi)$, which is the real part shifted a bit, so it seems to be nothing. But it still makes me uncomfortable to use $Ae^{i(\omega t - \psi)}$ in place of $A cos(\omega t - \psi)$ since I don't understand why adding $i sin(\omega t - \psi)$ (to get the rhs of euler's) is justifiable to begin with.
So how do I think of the complex exponential form of sinusoidal waves? When someone says "there's a sinusoid wave with amplitude $A$, frequency $\omega$, and phase shift $\psi$", can I safely use the complex form instead of the cosine form in my calculations, and just take the real part at the end of it?
