You may remember the following trigonometric identity for sinusoids:
In this way, a sinusoidal signal with arbitrary phase offset can be written:
where B and C are constants proportional to the amplitude A and the phase offset .
Thus, a sinusoid with arbitrary phase offset can be ``separated'' into sine and cosine terms. Said another way, a sinusoid with arbitrary phase offset can be completely reconstructed from properly scaled sine and cosine terms.
This result suggests a solution to the dot-product problem discussed above. That is, in order to fully detect a sinusoidal component in a given signal, we need to compute the dot-product with both a cosine and a sine term for each frequency (which produces results that correspond to the constants B and C above and from which we can also calculate the amplitude A and the phase offset ).