Quadruple Angle Formulas/Sine
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Theorem
- $\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$
where $\sin$ and $\cos$ denote sine and cosine respectively.
Corollary 1
For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
- $\dfrac {\sin 4 \theta} {\sin \theta} = 8 \cos^3 \theta - 4 \cos \theta$
where $\sin$ denotes sine and $\cos$ denotes cosine.
Corollary 2
For all $\theta$ such that $\theta \ne 0, \pm \pi, \pm 2 \pi \ldots$
- $\dfrac {\sin 4 \theta} {\sin \theta} = 2 \cos 3 \theta + 2 \cos \theta$
where $\sin$ denotes sine and $\cos$ denotes cosine.
Proof 1
\(\ds \map \sin {4 \theta}\) | \(=\) | \(\ds \map \sin {3 \theta + \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin 3 \theta \cos \theta + \cos 3 \theta \sin \theta\) | Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {3 \sin \theta - 4 \sin^3 \theta} \cos \theta + \paren {4 \cos^3 \theta - 3 \cos \theta} \sin \theta\) | Triple Angle Formula for Sine and Triple Angle Formula for Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sin \theta \cos \theta - 4 \sin^3 \theta \cos \theta + 4 \cos^3 \theta \sin \theta - 3 \cos \theta \sin \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 3 \sin \theta \cos \theta - 4 \sin^3 \theta \cos \theta + 4 \cos \theta \paren {1 - \sin^2 \theta} \sin \theta - 3 \cos \theta \sin \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta\) | multiplying out and gathering terms |
$\blacksquare$
Proof 2
We have:
\(\ds \cos 4 \theta + i \sin 4 \theta\) | \(=\) | \(\ds \paren {\cos \theta + i \sin \theta}^4\) | De Moivre's Formula | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\cos \theta}^4 + \binom 4 1 \paren {\cos \theta}^3 \paren {i \sin \theta} + \binom 4 2 \paren {\cos \theta}^2 \paren {i \sin \theta}^2\) | ||||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \binom 4 3 \paren {\cos \theta} \paren {i \sin \theta}^3 + \paren {i \sin \theta}^4\) | Binomial Theorem | ||||||||||
\(\ds \) | \(=\) | \(\ds \cos^4 \theta + 4 i \cos^3 \theta \sin \theta - 6 \cos^2 \theta \sin^2 \theta\) | substituting for binomial coefficients | |||||||||||
\(\ds \) | \(\) | \(\, \ds - \, \) | \(\ds 4 i \cos \theta \sin^3 \theta + \sin^4 \theta\) | and using $i^2 = -1$ | ||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \cos^4 \theta - 6 \cos^2 \theta \sin^2 \theta + \sin^4 \theta\) | |||||||||||
\(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds 4 i \cos^3 \theta \sin \theta - 4 i \cos \theta \sin^3 \theta\) | rearranging |
Hence:
\(\ds \sin 4 \theta\) | \(=\) | \(\ds 4 \cos^3 \theta \sin \theta - 4 \cos \theta \sin^3 \theta\) | equating imaginary parts in $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \cos \theta \paren {1 - \sin^2 \theta} \sin \theta - 4 \cos \theta \sin^3 \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \cos \theta \sin \theta - 8 \sin^3 \theta \cos \theta\) | multiplying out and gathering terms |
$\blacksquare$
Proof 3
\(\ds \sin {4 \theta}\) | \(=\) | \(\ds \map \sin {2 \times 2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sin 2 \theta \cos 2 \theta\) | Double Angle Formula for Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds 2 \paren {2 \sin \theta \cos \theta} \paren {\cos^2 \theta - \sin^2 \theta}\) | Double Angle Formula for Sine, Double Angle Formula for Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \sin \theta \cos^3 \theta - 4 \sin^3 \theta \cos \theta\) | Distributive Laws of Arithmetic | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \sin \theta \paren {1 - \sin^2 \theta} \cos \theta - 4 \sin^3 \theta \cos \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta\) | simplification |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.47$