Triple Angle Formulas/Cosine
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Theorem
- $\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$
where $\cos$ denotes cosine.
Example: $2 \cos 3 \theta + 1$
- $2 \cos 3 \theta + 1 = \paren {\cos \theta - \cos \dfrac {2 \pi} 9} \paren {\cos \theta - \cos \dfrac {4 \pi} 9} \paren {\cos \theta - \cos \dfrac {8 \pi} 9}$
Proof 1
| \(\ds \cos 3 \theta\) | \(=\) | \(\ds \cos \paren {2 \theta + \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta\) | Cosine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cos^2 \theta - \sin^2 \theta} \cos \theta - \paren {2 \sin \theta \cos \theta} \sin \theta\) | Double Angle Formula for Cosine and Double Angle Formula for Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^3 \theta - \sin^2 \theta \cos \theta - 2 \sin^2 \theta \cos \theta\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^3 \theta - \paren {1 - \cos^2 \theta} \cos \theta - 2 \paren {1 - \cos^2 \theta} \cos \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^3 \theta - \cos \theta + \cos^3 \theta - 2 \cos \theta + 2 \cos^3 \theta\) | multiplying out | |||||||||||
| \(\ds \) | \(=\) | \(\ds 4 \cos^3 \theta - 3 \cos \theta\) | gathering terms |
$\blacksquare$
Proof 2
We have:
| \(\ds \cos 3 \theta + i \sin 3 \theta\) | \(=\) | \(\ds \paren {\cos \theta + i \sin \theta}^3\) | De Moivre's Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\cos \theta}^3 + \binom 3 1 \paren {\cos \theta}^2 \paren {i \sin \theta}\) | Binomial Theorem | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds \binom 3 2 \paren {\cos \theta} \paren {i \sin \theta}^2 + \paren {i \sin \theta}^3\) | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^3 \theta + 3 i \cos^2 \theta \sin \theta + 3 i^2 \cos \theta \sin^2 \theta + i^3 \sin^3 \theta\) | substituting for binomial coefficients | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^3 \theta + 3 i \cos^2 \theta \sin \theta - 3 \cos \theta \sin^2 \theta - i \sin^3 \theta\) | $i^2 = -1$ | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \cos^3 \theta - 3 \cos \theta \sin^2 \theta\) | |||||||||||
| \(\ds \) | \(\) | \(\, \ds + \, \) | \(\ds i \paren {3 \cos^2 \theta \sin \theta - \sin^3 \theta}\) | rearranging |
Hence:
| \(\ds \cos 3 \theta\) | \(=\) | \(\ds \cos^3 \theta - 3 \cos \theta \sin^2 \theta\) | equating real parts in $(1)$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos^3 \theta - 3 \cos \theta \paren {1 - \cos^2 \theta}\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 4 \cos^3 \theta - 3 \cos \theta\) | multiplying out and gathering terms |
$\blacksquare$
Proof 3
| \(\ds \cos 3 \theta\) | \(=\) | \(\ds \cos \paren {2 \theta + \theta}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta\) | Cosine of Sum | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \cos^2 \theta - 1} \cos \theta - 2 \sin^2 \theta \cos \theta\) | Double Angle Formula for Cosine and Double Angle Formula for Sine | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \cos^2 \theta - 1 - 2 \paren {1 - 2 \cos^2 \theta} } \cos \theta\) | Sum of Squares of Sine and Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds 4 \cos^3 \theta - 3 \cos \theta\) | gathering terms |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.45$
- 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: De Moivre's Theorem: $90 \ \text{(b)}$