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.

$3 \cos x - 2 \sin x = \sqrt {13} \map \cos {x + \arctan \dfrac 2 3}$

Hence the greatest value of $3 \cos x - 2 \sin x$ is $\sqrt {13}$ which happens when $x = -\arctan \dfrac 2 3$.

Consider the equation:

$(1): \quad 3 \cos x - 2 \sin x = 1$

The solutions to $(1)$ between $0 \degrees$ and $360 \degrees$ are:

$\ds x$

$=$

$\ds 40 \degrees \, 20'$

$\ds x$

$=$

$\ds 252 \degrees \, 24'$

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 $(3)$.