Sextuple Angle Formulas/Sine/Corollary

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Theorem

$\sin 6 \theta = 6 \sin \theta \cos \theta - 32 \sin^3 \theta \cos \theta + 32 \sin^5 \theta \cos \theta$

where $\sin$ denotes sine and $\cos$ denotes cosine.

Proof

\(\ds \sin 6 \theta\)

\(=\)

\(\ds \paren {2 \cos \theta } \sin 5 \theta - \sin 4 \theta\)

Sine of Integer Multiple of Argument/Formulation 4

\(\ds \)

\(=\)

\(\ds \paren {2 \cos \theta } \paren { 5 \sin \theta - 20 \sin^3 \theta + 16 \sin^5 \theta } - \paren { 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta }\)

\(\ds \)

\(=\)

\(\ds \paren {10 - 4 } \sin \theta \cos \theta + \paren {-40 + 8} \sin^3 \theta \cos \theta + 32 \sin^5 \theta \cos \theta\)

Gathering terms

\(\ds \)

\(=\)

\(\ds 6 \sin \theta \cos \theta - 32 \sin^3 \theta \cos \theta + 32 \sin^5 \theta \cos \theta\)

$\blacksquare$