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I was studying Fourier transforms today, and I thought, what if you took the transform of an ordinary sine or cosine? Well, since they only have one frequency, shouldn't the transform have only one value? That is, a delta function centered at the angular frequency of the wave.

But when I was reading a random pdf I googled, it said that the transform of a cosine wave was a pair of delta functions, centered at plus/minus the frequency. I don't get it.

It also said that the transform of a constant, a straight horizontal line, is a delta function. I understand the opposite, that the transform of a delta function is a straight line, but why is the inverse true?

This kind of made me realize there is something fundamental I am not understanding. In the frequency space of the transform, are those frequencies of sines or cosines? That is, to get the original function would you add up all sine waves with those amplitudes (of the transform), or cosine waves? I think even this question is not correct (or clear).

I appreciate any clarification!

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# Fourier Transform, Delta Function

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