Auxiliary Angle/Examples/3 cos x minus 2 sin x equals 1
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Example of Auxiliary Angle
Consider the equation:
- $(1): \quad 3 \cos x - 2 \sin x = 1$
The solutions to $(1)$ between $0 \degrees$ and $360 \degrees$ are:
| \(\ds x\) | \(=\) | \(\ds 40 \degrees \, 20'\) | ||||||||||||
| \(\ds x\) | \(=\) | \(\ds 252 \degrees \, 24'\) |
Proof
From Multiple of Sine plus Multiple of Cosine: Cosine Form:
- $p \sin x + q \cos x = \sqrt {p^2 + q^2} \map \cos {x + \arctan \dfrac {-p} q}$
Hence the auxiliary angle form of $(1)$ is given by:
| \(\ds \sqrt {13} \cos {x + \alpha}\) | \(=\) | \(\ds 1\) | where $\alpha = \arctan \dfrac 2 3$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cos {x + \alpha}\) | \(=\) | \(\ds \dfrac 1 {\sqrt {13} }\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0.2773\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x + \alpha\) | \(=\) | \(\ds 73 \degrees \, 54'\) | |||||||||||
| \(\ds \) | \(\text {or}\) | \(\ds 286 \degrees \, 24'\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 40 \degrees \, 20'\) | as $\alpha = \arctan \dfrac 2 3 = 33 \degrees \, 42'$ | ||||||||||
| \(\ds \) | \(\text {or}\) | \(\ds 252 \degrees \, 24'\) |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Solution of equations: Example