I am trying to express this signal function as a sum of complex exponential signals
$$s(t) = 10 + 20 \cos(200\pi t+\pi 4) + 10 \cos(500\pi t).$$
I know that $e^{ix} = \cos(x) + i\sin(x)$ and the complex exponential signal takes the form $e^{ix}$.
So far, I've been able to write $10=10e^{i\cdot 0}$ How can I express the rest of the signal sum of signals' complex exponential forms if there are no imaginary sine parts in the equation?
You simply decompose each sine|cosine using:
$$e^{ix}=\cos(x)+i\sin(x) \\ e^{-ix}=\cos(x)-i\sin(x) \\ $$
$$\cos(x)={e^{ix}+e^{-ix} \over 2} \\ \sin(x)={e^{ix}-e^{-ix} \over 2i} \\ $$
Hence:
$$ s(t)=10+20\cos(200\pi t+4\pi)+10\cos(500\pi t) \\ =10+20{e^{i200\pi t+i4\pi}+e^{-i 200\pi t+i4\pi} \over 2}+10{e^{i 500\pi t}+e^{-i 500\pi t} \over 2} \\ =10+10e^{i200\pi t+i4\pi}+10e^{-i 200\pi t+i4\pi} +5e^{i 500\pi t}+5e^{-i 500\pi t} \\ $$