Quadruple Angle Formulas/Sine/Proof 3
Jump to navigation
Jump to search
Theorem
- $\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$
Proof
\(\ds \sin {4 \theta}\) | \(=\) | \(\ds \map \sin {2 \times 2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 \sin 2 \theta \cos 2 \theta\) | 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 |
$\blacksquare$