Definition:Auxiliary Angle
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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
- Multiple of Sine plus Multiple of Cosine, where it is shown what $\alpha$ is in terms of $p$ and $q$ for both forms $(2)$ and $(3)$.
- Results about auxiliary angles can be found here.
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: The auxiliary angle