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:


 * Auxiliary-angle-3cosx-2sinx.png

From the diagram:

where:
 * $\tan \alpha = \dfrac 2 3$

The result follows.