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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?

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I can think of one key counter example from my undergrad: Consider the energy density of an electromagnetic field

$$ u = \frac{1}{2}\mathbf{E}.\mathbf{D}+\frac{1}{2}\mathbf{B}.\mathbf{H}$$

Taking the time derivative:

$$\frac{du}{dt}= \frac{1}{2}\left( \dot{\mathbf{E}}.\mathbf{D}+\mathbf{D}.\dot{\mathbf{E}}+\dot{\mathbf{B}}.\mathbf{H}+\mathbf{B}.\dot{\mathbf{H}} \right)$$

Now suppose $\mathbf{E}$ and $\mathbf{D}$ vary sinusoidally. In calculating $\dot{\mathbf{E}}.\mathbf{D}$ you get different answers, when using the complex representation, depending on when you take the real part: $$\mathcal{R}\{\dot{\mathbf{E}}.\mathbf{D}\}\neq\mathcal{R}\{\dot{\mathbf{E}}\}.\mathcal{R}\{\mathbf{D}\}$$

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