In-phase and quadrature components

This page demonstrates how a sinusoid with arbitrary phase can be considered to be composed of in-phase and quadrature components, that is, two sinusoids with phases of 0° and 90°.

Since
sin(A + B) = sin(A) cos(B) + cos(A) sin(B) [reference]
it can be seen that
sin(ωt + φ) = a sin(ωt) + b cos(ωt)
where a = cos φ and b = sin φ.