Equation for Fourier transforms

Question

I am aware that the formula to transform a time-dependent function $f(t)$ into the frequency domain is$$\hat f(\omega)=\int_{-\infty}^{\infty}e^{i\omega t}f(t)dt.$$I understand the role that $e^{i\omega t}$ plays in this function, as it "filters out" all the components that alternate at an angular frequency other than $\omega$ (the value of its correlation function with any of these functions is $0$). However, it seems to me that any function with period $\omega$ would work equally well in the place of $e^{i\omega t}$. So what is the point of the imaginary component here? Why is the Fourier transform not just $\frac1{2\pi}\int_{-\infty}^{\infty}\cos(\omega t) \, f(t) \, dt$?