Here we multiply two signals with the same angular frequency but with different phases:
where denotes complex conjugate. For example,
Define , and make the algebra easier to follow by defining two phases:
Note that is the product of two binomials, which yield four terms:
When the two binomials are multiplied we obtain four terms. We group them according to whether they involve the sum or difference between the two phases, and , because whether it is a sum or difference affects the time-dependence as follows:
These terms can be grouped into real and imaginary parts, expressed in terms of the sine and cosine functions:
Graphs of current i, voltage v, and power p for an ac circuit with a phase shift between current and voltage.