If I had a complex signal such as: $s(t) = \cos(\pi t)*e^{-j\pi t}$, how do I go about simplifying this down so that I have an exponential which can let me calculate its magnitude and phase? This should be an easy thing, but I've forgotten a lot of concepts, which I hope someone can clear up.
Actually, if I needed to split this into its real and imaginary components, would it be easier to solve for the exponential form first, or directly rectangular?
Your signal is in polar form $s=re^{j\phi}$ where $r$ and $\phi$ are real. In such case, $|s|=r$ and $\measuredangle s=\phi$
Therefore, for the given signal:
$$|s(t)|=\cos(\pi t)$$ and $$\measuredangle s(t)=-\pi t $$