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

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

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

Proof

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

Explicit derivation illustrated below:

\(\ds \sin 5 \theta\)

\(=\)

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

Sine of Integer Multiple of Argument: Formulation 4

\(\ds \sin 3 \theta\)

\(=\)

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

Sine of Integer Multiple of Argument: Formulation 4

\(\ds \sin 5 \theta\)

\(=\)

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

Sine of Integer Multiple of Argument: Formulation 4

\(\ds \)

\(=\)

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

$\blacksquare$