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

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

$\cos 5 \theta = \dfrac 1 2 \paren {\paren {2 \cos \theta }^5 - 5 \paren {2 \cos \theta }^3 + 5 \paren {2 \cos \theta } }$

Proof

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

\(\ds \cos 5 \theta\)

\(=\)

\(\ds \frac 1 2 \paren {\paren {2 \cos \theta }^5 + \sum_{k \mathop \ge 1} \paren {-1 }^k \dfrac 5 k \dbinom {5 - \paren {k + 1 } } {k - 1} \paren {2 \cos \theta }^{5 - 2 k } }\)

Cosine of Integer Multiple of Argument: Formulation 1

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\paren {2 \cos \theta }^5 - 5 \paren {2 \cos \theta }^{5 - 2} + \dfrac 5 2 \dbinom {5 - 3} 1 \paren {2 \cos \theta }^{5 - 4} }\)

\(\ds \)

\(=\)

\(\ds \frac 1 2 \paren {\paren {2 \cos \theta }^5 - 5 \paren {2 \cos \theta }^3 + \dfrac 5 2 \dbinom 2 1 \paren {2 \cos \theta } }\)

\(\ds \)

\(=\)

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

$\blacksquare$