Sine of Integer Multiple of Argument/Formulation 5/Examples/Sine of Sextuple Angle

Example of Use of Sine of Integer Multiple of Argument/Formulation 5

$\sin 6 \theta = 2 \cos \theta \paren {\sin 5 \theta - \sin 3 \theta + \sin \theta}$

Proof

Follows directly from the Sine of Integer Multiple of Argument: Formulation 5:

Explicit derivation illustrated below:

\(\ds \sin 6 \theta\)

\(=\)

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

Sine of Integer Multiple of Argument: Formulation 4

\(\ds \sin 4 \theta\)

\(=\)

\(\ds \paren {2 \cos \theta} \sin 3 \theta - \sin 2 \theta\)

Sine of Integer Multiple of Argument: Formulation 4

\(\ds \sin 6 \theta\)

\(=\)

\(\ds \paren {2 \cos \theta} \sin 5 \theta - \paren {\paren {2 \cos \theta} \sin 3 \theta - \sin 2 \theta}\)

\(\ds \)

\(=\)

\(\ds 2 \cos \theta \paren {\sin 5 \theta - \sin 3 \theta + \sin \theta}\)

Double Angle Formula for Sine

$\blacksquare$