Multiple of Sine plus Multiple of Cosine

Theorem

 * $p \sin x + q \cos x = \sqrt {p^2 + q^2} \cos \left({x + \arctan \dfrac {-p} q}\right)$

Proof
Let it be assumed that $p \sin x + q \cos x$ can be expressed in the form $M \cos \left({x + \phi}\right)$.

Then:

Then:

Hence the result.