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

Example of Use of Cosine of Integer Multiple of Argument: Formulation $5$

$\map \cos {6 \theta } = -1 + 2 \cos \theta \paren {\cos 5 \theta - \cos 3 \theta + \cos \theta }$

Proof

Follows directly from the Cosine of Integer Multiple of Argument: Formulation $5$:

Explicit derivation illustrated below:

\(\ds \map \cos {6 \theta}\)

\(=\)

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

Cosine of Integer Multiple of Argument: Formulation $4$

\(\ds \map \cos {4 \theta}\)

\(=\)

\(\ds 2 \cos \theta \map \cos {3 \theta} - \map \cos {2 \theta}\)

Cosine of Integer Multiple of Argument: Formulation $4$

\(\ds \map \cos {6 \theta}\)

\(=\)

\(\ds 2 \cos \theta \paren { \map \cos {5 \theta} - \map \cos {3 \theta} + \cos \theta} - 1\)

Double Angle Formula for Cosine: Corollary $1$

\(\ds \)

\(=\)

\(\ds -1 + 2 \cos \theta \paren {\cos 5 \theta - \cos 3 \theta + \cos \theta}\)

$\blacksquare$