Triple Angle Formulas/Cosine/Proof 1
Jump to navigation
Jump to search
Theorem
- $\cos 3 \theta = 4 \cos^3 \theta - 3 \cos \theta$
Proof
\(\ds \cos 3 \theta\) | \(=\) | \(\ds \cos \paren {2 \theta + \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos 2 \theta \cos \theta - \sin 2 \theta \sin \theta\) | 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 |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Two more useful formulae