Cosine of Integer Multiple of Argument/Formulation 5
Theorem
For $n \in \Z_{>0}$:
\(\ds \cos n \theta\) | \(=\) | \(\ds \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + \paren {2 \cos \theta} \paren {\map \cos {\paren {n - 1} \theta} - \map \cos {\paren {n - 3} \theta} + \map \cos {\paren {n - 5} \theta} - \cdots}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }\) |
Proof
The proof proceeds by induction.
For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:
- $\ds \cos n \theta = \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }$
Basis for the Induction
$\map P 1$ is the case:
\(\ds \cos \theta\) | \(=\) | \(\ds \cos \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\frac {\paren {1 + 1} \pi} 2} + \paren {\sin \frac \pi 2} \paren {\cos \theta} + 2 \cos \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \cos {\paren {1 - 0} \theta} }\) |
So $\map P 1$ is seen to hold.
$\map P 2$ is the case:
\(\ds \cos 2 \theta\) | \(=\) | \(\ds 2 \cos^2 \theta - 1\) | Double Angle Formula for Cosine: Corollary $2$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\frac {\paren {2 + 1} \pi} 2} + \paren {\sin \frac {2 \pi} 2} \paren {\cos \theta} + 2 \cos \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \cos {\paren {2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \cos {\paren {2 - 1} \theta} }\) |
So $\map P 2$ is also seen to hold.
$\map P 3$ is the case:
\(\ds \cos 3 \theta\) | \(=\) | \(\ds 2 \cos \theta \cos 2 \theta - \cos \theta\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\frac {\paren {3 + 1} \pi} 2} + \paren {\sin \frac {3 \pi} 2} \paren {\cos \theta} + 2 \cos \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \cos {\paren {3 - 0} \theta} + \paren {\sin \frac \pi 2} \map \cos {\paren {3 - 1} \theta} + \paren {\sin \frac {2 \pi} 2} \map \cos {\paren {3 - 2} \theta} }\) |
So $\map P 3$ is also seen to hold.
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P n$ is true, where $n > 3$ and $n$ even, then it logically follows that $\map P {n + 2}$ is true.
We also need to show that, if $\map P n$ is true, where $n > 3$ and $n$ odd, then it logically follows that $\map P {n + 2}$ is true.
So this is our induction hypothesis:
- $\ds \map \cos {n \theta} = \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \paren {\cos \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }$
from which we are to show:
- $\ds \map \cos {\paren {n + 2} \theta} = \map \sin {\frac {\paren {n + 3} \pi} 2} + \paren {\sin \frac {\paren {n + 2} \pi} 2} \paren {\cos \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n + 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n + 2 - k} \theta} }$
Induction Step
This is our induction step:
For $n$ even or for $n$ odd: (Identical argument)
\(\ds \map \cos {\paren {n + 2} \theta}\) | \(=\) | \(\ds \paren {2 \cos \theta} \map \cos {\paren {n + 1} \theta} - \map \cos {\paren n \theta}\) | Cosine of Integer Multiple of Argument: Formulation 4 | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {2 \cos \theta} \map \cos {\paren {n + 1} \theta} - \map \sin {\frac {\paren {n + 1} \pi} 2} - \paren {\sin \frac {n \pi} 2} \cos \theta - 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\map \sin {\frac {\paren {n + 1} \pi} 2} - \paren {\sin \frac {n \pi} 2} \cos \theta + \paren {2 \cos \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \cos {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \cos {\paren {n + 2 - 1} \theta} } - 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }\) | rearranging terms and adding zero | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\frac {\paren {n + 3} \pi} 2} + \paren {\map \sin {\frac {n \pi} 2 + \pi} } \paren {\cos \theta} + \paren {2 \cos \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \cos {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \cos {\paren {n + 2 - 1} \theta} } - 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }\) | Sine of Angle plus Integer Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\frac {\paren {n + 3} \pi} 2} + \paren {\sin \frac {\paren {n + 2} \pi} 2} \paren {\cos \theta} + \paren {2 \cos \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \cos {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \cos {\paren {n + 2 - 1} \theta} } + 2 \cos \theta \paren {\sum_{k \mathop = 2}^{n + 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n + 2 - k} \theta} }\) | Translation of Index Variable of Summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \map \sin {\frac {\paren {n + 3} \pi} 2} + \paren {\sin \frac {\paren {n + 2} \pi} 2} \paren {\cos \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n + 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n + 2 - k} \theta} }\) |
The result follows by the Principle of Mathematical Induction.
Therefore:
- $\ds \forall n \in \Z_{>0}: \cos n \theta = \map \sin {\frac {\paren {n + 1} \pi} 2} + \paren {\sin \frac {n \pi} 2} \cos \theta + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n - 1} \paren {\sin \frac {k \pi} 2} \map \cos {\paren {n - k} \theta} }$
$\blacksquare$
Examples
Cosine of Quintuple Angle
- $\cos 5 \theta = \cos \theta + 2 \cos \theta \paren {\cos 4 \theta - \cos 2 \theta}$
Cosine of Sextuple Angle
- $\map \cos {6 \theta } = -1 + 2 \cos \theta \paren {\cos 5 \theta - \cos 3 \theta + \cos \theta }$