I'm trying to find the Fourier transform of
$\ f(t) = 25 \cdot \sin(8t + 5) + 13 \cdot \cos(7t-3)$.
I understand that this can somehow be done with Dirac's delta function as a help, I understand the definition of the delta function but I don't know how to actually apply it in this case. If anyone could give me some pointers I'd be grateful!
Let's define the Fourier transform as \begin{align} \mathcal{F}\{f(t)\}=F(\omega)=\int_{-\infty }^{\infty }f(t)e^{-i\omega t}\,\mathrm dt \end{align} Observing that \begin{align} \mathcal{F}\{\cos(at)\}&=\pi\left[\delta(\omega-a)+\delta(\omega+a)\right]\\ \mathcal{F}\{\sin(at)\}&=-i\pi\left[\delta(\omega-a)-\delta(\omega+a)\right] \end{align} we have \begin{align} F(\omega)&=-25\,\mathrm e^{i5\omega}i\pi\left[\delta(\omega-8)-\delta(\omega+8)\right]+ 13\,\mathrm e^{-i3\omega}\,\pi\left[\delta(\omega-7)+\delta(\omega+7)\right] \end{align}