Fourier Transform of $\cos(3\pi t+2)$

Question

How would one go about solving the Fourier Transform of

$$f(t) = cos(3\pi t+2)$$

Is expanding the trig function into $cos(A+B) = cos(A)cos(B) - sin(A)sin(B)$ correct?

That is:

$F(w) = \pi \times cos(2) \big[\delta(w + 3\pi) + \delta(w - 3\pi) \big] - \cfrac{\pi \times sin(2)}{j}\big[\delta(w + 3\pi) - \delta(w - 3\pi) \big]$

If so, is there another method to solving this?

Answer 1

I would suggest taking advantage of the properties of the Fourier Transform, specifically Time scaling and Translation.

Checkout https://en.wikipedia.org/wiki/Fourier_transform#Properties_of_the_Fourier_transform

Answer 2

My advice would be to look to @M. Jaj answer, but you can also notice:

$$\mathcal{F}_{t}\left[a\cos(bt+c)\right]_{(\omega)}=\sqrt{\frac{\pi}{2}}ae^{-ci}\left(\delta(\omega-b)+e^{2ci}\delta(b+\omega)\right)$$

So, when $a=1,b=3\pi$ and $c=2$:

$$\mathcal{F}_{t}\left[\cos(3\pi t+2)\right]_{(\omega)}=\sqrt{\frac{\pi}{2}}e^{-2i}\left(\delta(\omega-3\pi)+e^{4i}\delta(3\pi+\omega)\right)$$