In-Phase and Phase-Quadrature Components

Every sinusoid can be expressed as the sum of a sine and cosine function, or equivalently, an ``in-phase'' and ``phase-quadrature'' component.

Using the trigonometric identity²

$$\sin(a\pm b)=\sin(a)\cos(b)\pm\cos(a)\sin(b),$$

we see that

$$A\sin(\omega_0 t + \phi) = A\sin(\phi + \omega_0 t)$$

$$= [A \sin\phi]\cos\omega_0 t + [A\cos\phi]\sin\omega_0 t$$

$$= B\cos\omega_0 t + C\sin\omega_0 t,$$

where the amplitude $A$ is given by

$$A = \sqrt{B^2 + C^2}, \qquad$$

and the phase $\phi$ is given by

$$\phi = \tan^{-1}\left(\frac{B}{C}\right).$$