# Triple Angle Formulas/Sine/Proof 1

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## Theorem

$\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta$

## Proof

 $\displaystyle \sin 3 \theta$ $=$ $\displaystyle \map \sin {2 \theta + \theta}$ $\displaystyle$ $=$ $\displaystyle \sin 2 \theta \cos \theta + \cos 2 \theta \sin \theta$ Sine of Sum $\displaystyle$ $=$ $\displaystyle \paren {2 \sin \theta \cos \theta} \cos \theta + \paren {\cos^2 \theta - \sin^2 \theta} \sin \theta$ Double Angle Formula for Sine and Double Angle Formula for Cosine $\displaystyle$ $=$ $\displaystyle 2 \sin \theta \cos^2 \theta + \cos^2 \theta \sin \theta - \sin^3 \theta$ $\displaystyle$ $=$ $\displaystyle 2 \sin \theta \paren {1 - \sin^2 \theta} + \paren {1 - \sin^2 \theta} \sin \theta - \sin^3 \theta$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle 2 \sin \theta - 2 \sin^3 \theta + \sin \theta - \sin^3 \theta - \sin^3 \theta$ multiplying out $\displaystyle$ $=$ $\displaystyle 3 \sin \theta - 4 \sin^3 \theta$ gathering terms

$\blacksquare$