Quintuple Angle Formulas/Sine
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Theorem
- $\sin 5 \theta = 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta$
where $\sin$ denotes sine.
Corollary
For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
- $\dfrac {\sin 5 \theta} {\sin \theta} = 16 \cos^4 \theta - 12 \cos^2 \theta + 1$
where $\sin$ denotes sine and $\cos$ denotes cosine.
Proof 1
| \(\ds \sin 5 \theta\) | \(=\) | \(\ds \map \sin {3 \theta + 2 \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \sin 3 \theta \cos 2 \theta + \cos 3 \theta \sin 2 \theta\) | Sine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \cos 2 \theta + \paren {4 \cos^3 \theta - 3 \cos \theta} \sin 2 \theta\) | Triple Angle Formulas | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \paren {\cos^2 \theta - \sin^2 \theta} + \paren {4 \cos^3 \theta - 3 \cos \theta} 2 \sin \theta \cos \theta\) | Double Angle Formulas | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \paren {\cos^2 \theta - \sin^2 \theta} + 8 \cos^4 \theta \sin \theta - 6 \cos^2 \theta \sin \theta\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \paren {1 - 2 \sin^2 \theta} + 8 \paren {1 - \sin^2 \theta}^2 \sin \theta - 6 \paren {1 - \sin^2 \theta} \sin \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 5 \sin \theta - 20 \sin^3 \theta + 16 \sin ^5 \theta\) | multiplying out and gathering terms |
$\blacksquare$
Proof 2
We have:
| \(\ds \cos 5 \theta + i \sin 5 \theta\) | \(=\) | \(\ds \paren {\cos \theta + i \sin \theta}^5\) | De Moivre's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cos \theta}^5 + \binom 5 1 \paren {\cos \theta}^4 \paren {i \sin \theta} + \binom 5 2 \paren {\cos \theta}^3 \paren {i \sin \theta}^2\) | ||||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \binom 5 3 \paren {\cos \theta}^2 \paren {i \sin \theta}^3 + \binom 5 4 \paren {\cos \theta} \paren {i \sin \theta}^4 + \paren {i \sin \theta}^5\) | Binomial Theorem | ||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^5 \theta + 5 i \cos^4 \theta \sin \theta - 10 \cos^3 \theta \sin^2 \theta\) | substituting for binomial coefficients | |||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 10 i \cos^2 \theta \sin^3 \theta + 5 \cos \theta \sin^4 \theta + i \sin^5 \theta\) | and using $i^2 = -1$ | ||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \cos^5 \theta - 10 \cos^3 \theta \sin^2 \theta + 5 \cos \theta \sin^4 \theta\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds i \paren {5 \cos^4 \theta \sin \theta - 10 \cos^2 \theta \sin^3 \theta + \sin^5 \theta}\) | rearranging |
Hence:
| \(\ds \sin 5 \theta\) | \(=\) | \(\ds 5 \cos^4 \theta \sin \theta - 10 \cos^2 \theta \sin^3 \theta + \sin^5 \theta\) | equating imaginary parts in $(1)$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds 5 \paren {1 - \sin^2 \theta}^2 \sin \theta - 10 \paren {1 - \sin^2 \theta} \sin^3 \theta + \sin^5 \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta\) | multiplying out and gathering terms |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.50$