Spectrum of a sinusoid multiplied by the cardinal sine

Question

I think it'd be easier to understand if i gave an example; let's say we have this function:

$$\cos(200 \pi t) \cdot \mbox{sinc} (5t)$$

What would the correct procedure to calculate the spectrum of the signal be?

Usually if it were a simpler function like: $$cos(10 \pi t) \cdot sin(5 \pi t)$$

i would use Euler's formula and i'd get:

$$\frac{e^{i10\pi t} + e^{-i10\pi t}}{2} \cdot \frac{e^{i5\pi t} - e^{-i5\pi t}}{2i} $$

then elaborate and plot the phase/magnitude schema. Should i follow a similar approach? Otherwise, what would the right approach be?

Thanks!

Answer 1

Hint

In both cases you can use the convolution theorem:

$$\mathcal{F}\{f(t).g(t)\}=\mathcal{F}\{g(t)\}*\mathcal{F}\{g(t)\}$$