Fourier transform of a sinc function

I'm trying to show the fourier transform of a since function:

I can't figure out how to show this. I could just work through the integral of the fourier transform, but that seems difficult for me... I think I could show it more easily using the duality property. i.e, finding the fourier transform of a rectangle function, then applying the duality property to show that transform the other way around. But I can't figure out what the rectangle function should be!

Can anyone help out or give some hints? Am I on the right track? Thanks!

Re: Fourier transform of a sinc function

OK, I've been working on this... I've got a few approaches. The first is this:

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From theory, we know that the fourier transform of a rectangle function is a sinc:

So, if the fourier transform of is , using the symmetry property (duality):

We can get

From the similarity property of the fourier transform, we get

so, scaling t in by a gives

so to find , substitute for a in the previous line

To find 2sinc(st) from the original question, multiply both sides by two using the linearity property.

So, if the fourier transform of is , using the symmetry property (duality):

We can get

From the similarity property of the fourier transform, we get

so, scaling t in by a gives

so to find , substitute for a in the previous line

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This works for me. Do you think this would suffice to answer the question posed above?

The other approach.. I'll split into a seperate post later on, got to get to work!