Multiple of Sine plus Multiple of Cosine/Sine Form

Theorem

 * $p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \sin {x + \arctan \dfrac q p}$

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

Then:

Then:

Hence the result.