Euler's Formula Conversion with coefficients

Question

If I have an equation such as $x(t) = \displaystyle \sum_{n=1}^N \left( a_n \cos(\omega_nt) + b_n \sin(\omega_n t) \right)$, how do I convert it to a sum of complex exponentials? In other words what do I do with the coefficients in front of the sine and cosine to turn them into the coefficient of each complex exponential.

Answer 1

Write $\displaystyle \cos(w_n t) = \frac{e^{i w_n t} + e^{-i w_n t}}{2}$ and $\displaystyle \sin(w_n t) = \frac{e^{i w_n t} - e^{-i w_n t}}{2i}$ and rearrange to convert it to a sum of complex exponentials.