I've been doing some thinking lately on the Fourier Transform and believe I had a fundamental misconception.
I had always visualized the Fourier Transform as an iterative process in which a sinusoidal function (say cosine) was dot multiplied by a signal of interest. I then believed that if the frequency being multiplied by the signal (with no regard of the phase) was contained within the signal of interest, there would be some 'amount' generated by the inner product operation. However, if the signal did not contain the frequency it was being multiplied by, the inner product would yield zero.
However, I'm starting to think this was a naive perspective ever since noticing that the inner product of cosine and sine of the same frequency seems to always yield an inner product of zero..
Thus, if two sinusoidal functions are completely out of phase, will their inner product always equal zero? What happens if the two signals are just slightly out of phase? Will the inner product still be zero?
$$ \int_0^{2 \pi} \sin x \sin (x + c) dx = \pi \cos c $$