Sine of Integer Multiple of Argument/Formulation 7/Examples/Sine of Quintuple Angle

Example of Use of Sine of Integer Multiple of Argument/Formulation 7

$\sin 5 \theta = \sin \theta + 2 \sin \theta \paren {\cos 4 \theta + \cos 2 \theta}$

Proof

Follows directly from the Sine of Integer Multiple of Argument: Formulation 7:

Explicit derivation illustrated below:

\(\ds \sin 5 \theta\)

\(=\)

\(\ds \paren {2 \sin \theta} \cos 4 \theta + \sin 3 \theta\)

Sine of Integer Multiple of Argument: Formulation 6

\(\ds \sin 3 \theta\)

\(=\)

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

Sine of Integer Multiple of Argument: Formulation 6

\(\ds \leadsto \ \ \)

\(\ds \sin 5 \theta\)

\(=\)

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

\(\ds \)

\(=\)

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

$\blacksquare$