- You may remember the following trigonometric identity for sinusoids:

(4) - 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 ).

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