Sine plus Sine/Examples/Sine 2 x plus Sine 5 x equals Sine 4 x
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Examples of Use of Sine plus Sine
The equation
- $\sin 3 x + \sin 5 x = \sin 4 x$
has the general solution:
- $\set {\dfrac {n \pi} 4 : n \in \Z} \cup \set {2 n \pi \pm \dfrac \pi 3: n \in \Z}$
Proof
| \(\ds \sin 3 x + \sin 5 x\) | \(=\) | \(\ds \sin 4 x\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 2 \sin 4 x \cos x\) | \(=\) | \(\ds \sin 4 x\) | Sine plus Sine | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin 4 x \paren {2 \cos x - 1}\) | \(=\) | \(\ds 0\) | simplifying | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \sin 4 x\) | \(=\) | \(\ds 0\) | equating factors | ||||||||||
| \(\ds \) | \(=\) | \(\ds \sin 0\) | Sine of Zero is Zero | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \dfrac {n \pi} 4\) | Solutions of $\sin x = \sin a$ | ||||||||||
| \(\, \ds \lor \, \) | \(\ds 2 \cos x - 1\) | \(=\) | \(\ds 0\) | equating factors | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \cos x\) | \(=\) | \(\ds \dfrac 1 2\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \dfrac \pi 3\) | Cosine of $60 \degrees$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds 2 n \pi \pm \dfrac \pi 3\) | Solutions of $\cos x = \cos a$ |
Hence the result.
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Solution of equations: Example $3$.