In the Fourier transform of a function, we want to see if a function matches closely with a sinusoid or cosine with a frequency, $\omega$. Now let's say the function we're doing the transform of is given by:
$$f(t) = \cos(\gamma t+\phi)$$ And the Fourier transform is given by:
$$F(\omega) = \int\limits_{-\infty}^\infty f(t) e^{-i \omega t} dt$$
I have a hard time seeing how the phase, $\phi$ won't cause trouble at $\omega = \gamma$.
Expanding $e^{-i \omega t} = \cos(\omega t) - i \sin(\omega t)$, both the cos and sin terms should be of too much significance in the integral when $\omega = \gamma$ and $\phi$ is some random number.
Is Euler's formula supposed to somehow account for the phase? If so, what am I missing?