# 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.

## Examples

### $3 \cos x$ minus $2 \sin x$

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

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

### Solutions to $3 \cos x - 2 \sin x = 1$

Consider the equation:

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

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

 $\ds x$ $=$ $\ds 40 \degrees \, 20'$ $\ds x$ $=$ $\ds 252 \degrees \, 24'$

## Also see

• Results about auxiliary angles can be found here.