Auxiliary Angle/Examples/3 cos x minus 2 sin x

Example of Auxiliary Angle

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

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

Proof

From Multiple of Sine plus Multiple of Cosine: Cosine Form:

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

From the diagram:

\(\ds 3\)

\(=\)

\(\ds \sqrt {13} \cos \alpha\)

\(\ds 2\)

\(=\)

\(\ds \sqrt {13} \sin \alpha\)

\(\ds \leadsto \ \ \)

\(\ds 3 \cos x - 2 \sin x\)

\(=\)

\(\ds \sqrt {13} \paren {\cos \alpha \cos x - \sin \alpha \sin x}\)

\(\ds \)

\(=\)

\(\ds \sqrt {13} \map \cos {x + \alpha}\)

Cosine of Sum

where:

$\tan \alpha = \dfrac 2 3$

The result follows.

$\blacksquare$

Sources

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