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

Example of Use of Cosine of Integer Multiple of Argument: Formulation 7

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

Proof

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

Explicit derivation illustrated below:

\(\ds \map \cos {5 \theta}\)

\(=\)

\(\ds -2 \sin \theta \sin 4 \theta + \cos 3 \theta\)

Cosine of Integer Multiple of Argument: Formulation 6

\(\ds \map \cos {3 \theta}\)

\(=\)

\(\ds -2 \sin \theta \sin 2 \theta + \cos \theta\)

Cosine of Integer Multiple of Argument: Formulation 6

\(\ds \cos 5 \theta\)

\(=\)

\(\ds -2 \sin \theta \sin 4 \theta - 2 \sin \theta \sin 2 \theta + \cos \theta\)

\(\ds \)

\(=\)

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

$\blacksquare$