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

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

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

Proof

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

Explicit derivation illustrated below:

\(\ds \cos 5 \theta\)

\(=\)

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

Cosine of Integer Multiple of Argument: Formulation 4

\(\ds \cos 3 \theta\)

\(=\)

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

Cosine of Integer Multiple of Argument: Formulation 4

\(\ds \leadsto \ \ \)

\(\ds \cos 5 \theta\)

\(=\)

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

\(\ds \)

\(=\)

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

$\blacksquare$