Sextuple Angle Formulas/Sine
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Theorem
- $\dfrac {\sin 6 \theta} {\sin \theta} = 32 \cos^5 \theta - 32 \cos^3 \theta + 6 \cos \theta$
where $\cos$ denotes cosine and $\sin$ denotes sine.
Corollary
- $\sin 6 \theta = 6 \sin \theta \cos \theta - 32 \sin^3 \theta \cos \theta + 32 \sin^5 \theta \cos \theta$
Proof
| \(\ds \cos 6 \theta + i \sin 6 \theta\) | \(=\) | \(\ds \paren {\cos \theta + i \sin \theta}^6\) | De Moivre's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cos \theta}^6 + \binom 6 1 \paren {\cos \theta}^5 \paren {i \sin \theta} + \binom 6 2 \paren {\cos \theta}^4 \paren {i \sin \theta}^2\) | Binomial Theorem | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \binom 6 3 \paren {\cos \theta}^3 \paren {i \sin \theta}^3 + \binom 6 4 \paren {\cos \theta}^2 \paren {i \sin \theta}^4 + \binom 6 5 \paren {\cos \theta} \paren {i \sin \theta}^5 + \paren {i \sin \theta}^6\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^6 \theta + 6 i \cos^5 \theta \sin \theta - 15 \cos^4 \sin^2 \theta\) | substituting for binomial coefficients | |||||||||||
| \(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 20 i \cos^3 \theta \sin^3 \theta + 15 \cos^2 \theta \sin^4 \theta + 6 i \cos \theta \sin^5 \theta - \sin^6 \theta\) | and using $i^2 = -1$ | ||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \cos^6 \theta - 15 \cos^4 \sin^2 \theta + 15 \cos^2 \theta \sin^4 \theta - \sin^6 \theta\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds i \paren {6 \cos^5 \theta \sin \theta - 20 \cos^3 \theta \sin^3 \theta + 6 \cos \theta \sin^5 \theta}\) | rearranging |
Hence:
| \(\ds \sin 6 \theta\) | \(=\) | \(\ds 6 \cos^5 \theta \sin \theta - 20 \cos^3 \theta \sin^\theta + 6 \cos \theta \sin^5 \theta\) | equating imaginary parts in $(1)$ | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds \dfrac {\map \sin {6 \theta} } {\sin \theta}\) | \(=\) | \(\ds 6 \cos^5 \theta - 20 \cos^3 \theta \sin^2 \theta + 6 \cos \theta \sin^4 \theta\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds 6 \cos^5 \theta - 20 \cos^3 \theta \paren {1 - \cos^2 \theta} + 6 \cos \theta \paren {1 - \cos^2 \theta}^2\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 32 \cos^5 \theta - 32 \cos^3 \theta + 6 \cos \theta\) | multiplying out and gathering terms |
$\blacksquare$
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 2$. Geometrical Representations: Exercise $9$