Multiple of Sine plus Multiple of Cosine

Theorem

Cosine Form

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

Sine Form

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

Also see

• Definition:Auxiliary Angle

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: The auxiliary angle