Cosine of Integer Multiple of Argument/Formulation 1/Examples/Cosine of Sextuple Angle

Example of Use of Cosine of Integer Multiple of Argument/Formulation 1

$\cos 6 \theta = \dfrac 1 2 \paren {\paren {2 \cos \theta }^6 - 6 \paren {2 \cos \theta }^4 + 9 \paren {2 \cos \theta }^2 - 2 }$

Proof

Follows directly from the Cosine of Integer Multiple of Argument: Formulation 1:

\(\ds \cos 6 \theta\)

\(=\)

\(\ds \frac 1 2 \paren {\paren {2 \cos \theta }^6 + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac 6 k \dbinom {6 - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{6 - 2 k } }\)

Cosine of Integer Multiple of Argument: Formulation 1

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\paren {2 \cos \theta }^6 - 6 \dbinom {6 - 2} 0 \paren {2 \cos \theta }^{6 - 2} + \dfrac 6 2 \dbinom {6 - 3} 1 \paren {2 \cos \theta }^{6 - 4} - \dfrac 6 3 \dbinom {6 - 4} 2 \paren {2 \cos \theta }^{6 - 6} }\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\paren {2 \cos \theta }^6 - 6 \paren {2 \cos \theta }^4 + 3 \dbinom 3 1 \paren {2 \cos \theta }^2 - 2 }\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\paren {2 \cos \theta }^6 - 6 \paren {2 \cos \theta }^4 + 9 \paren {2 \cos \theta }^2 - 2 }\)

$\blacksquare$