Definition:Auxiliary Angle

Definition
Consider the expression:


 * $(1): \quad p \sin x + q \cos x$

where $x \in \R$.

Let $(1)$ be expressed in the form:
 * $(2): \quad R \map \cos {x + \alpha}$

or:
 * $(3): \quad R \map \sin {x + \alpha}$

The angle $\alpha$ is known as the auxiliary angle of either $(2)$ or $(3)$ as appropriate.

Also see

 * Multiple of Sine plus Multiple of Cosine, where it is shown what $\alpha$ is in terms of $p$ and $q$ for both forms $(2)$ and $(3)$.