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