I attempted to take the Fourier series as follows:
f(t) = (5+rect(t/4))cos(60pi*t)
= 5cos(60pi*t) + rect(t/4)cos(60pi*t)
I used the transformation for cos(w0t) and rect(t/4) to come up with
F(w) = 5pi(delta(w-60pi) + delta(w+60pi)) + sinc(2w) * pi(delta(w-60pi) + delta(w+60pi))
(note that asterisk in the last line indicates convolution not multiplication)

But now I can't figure out how to convolve sinc(2w) and delta(w-60pi) since they have different coefficients in front of w. I know that f(t) * delta(t-t0) = f(t-t0), but that formula doesn't seem to apply here...

I thought that maybe in order to simplify it I could break sinc(2w) into sin(w)cos(w)/w but that seems like it'll leave me with an insanely complex solution to try to do the inverse fourier transform on later in the problem. Does anyone have any ideas?