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

$\blacksquare$