after a few hours of trying to solve this problem, I've resorted to asking online.
"Given that $u\hat(\xi,t)$ is the Fourier Transform of $u(x,t)$ for any each fixed t, and $f\hat(\xi)$ is the Fourier Transform of $f(x)$. Determine the inverse Fourier Transform, $u(x,t)$ of
$u\hat(\xi,t)=f\hat(\xi)\bullet cos(\xi ct)$
You may use the translation and convolution properties of Fourier Transforms."
I assume that I somehow need to find $g(x)$ such that $g\hat(\xi)=cos(\xi ct)$, but the for this function, the Inverse Fourier Equation does not seem to converge.
Thanks in advance for the help!