I'm trying to find the Exponential Fourier coefficients for the following function $$ f(t)=\cos(t + \frac{\pi}{3})$$
From the definition i get that
$$ c_k = \frac{1}{T}\int_{p}\cos(t + \frac{\pi}{3})e^{-ik{\Omega}t}dt $$
T in this case is $2\pi$ so i get that $\Omega = 1$ and
$$ c_k = \frac{1}{T}\int_{p}\cos(t + \frac{\pi}{3})e^{-ikt}dt $$
If i now use the fact that $ e^{-ikt} = \cos(kt) - i\sin(kt)$ i get that
$$c_k = \frac{1}{2\pi}\int^{2\pi}_{0}\cos(t + \frac{\pi}{3})\cos(kt) dt - \frac{i}{2\pi}\int^{2\pi}_{0}\cos(t + \frac{\pi}{3})\sin(kt) dt $$
Since the integrand in the second integral is odd it should be $0$. Im now left with
$$c_k = \frac{1}{2\pi}\int^{2\pi}_{0}\cos(t + \frac{\pi}{3})\cos(kt) dt $$ and i cant seem to solve this integral. Moreover i have been told that you can find the Exponential Fourier coefficients for this function without solving any integral, but i can't see why. Any hints would be greatly appreciated.