Sine of Integer Multiple of Argument/Formulation 5

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Theorem

For $n \in \Z_{>0}$:

\(\ds \sin n \theta\) \(=\) \(\ds \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\map \sin {\paren {n - 1} \theta} - \map \sin {\paren {n - 3} \theta} + \map \sin {\paren {n - 5} \theta} - \cdots}\)
\(\ds \) \(=\) \(\ds \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\)


Proof

The proof proceeds by induction.


For all $n \in \Z_{>0}$, let $\map P n$ be the proposition:

$\ds \sin n \theta = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$


Basis for the Induction

$\map P 1$ is the case:

\(\ds \sin \theta\) \(=\) \(\ds \sin \theta\)
\(\ds \) \(=\) \(\ds \paren {\sin \frac \pi 2} \paren {\sin \theta} + 2 \cos \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {1 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {1 - 1} \theta} }\)

So $\map P 1$ is seen to hold.


$\map P 2$ is the case:

\(\ds \sin 2 \theta\) \(=\) \(\ds 2 \sin \theta \cos \theta\) Double Angle Formula for Sine
\(\ds \) \(=\) \(\ds \paren {\sin \frac {2 \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {2 - 1} \theta} + \paren {\sin \frac {2 \pi} 2} \map \sin {\paren {2 - 2} \theta} }\)

So $\map P 2$ is also seen to hold.


$\map P 3$ is the case:

\(\ds \sin 3 \theta\) \(=\) \(\ds 2 \cos \theta \sin 2 \theta - \sin \theta\) Sine of Integer Multiple of Argument: Formulation 4
\(\ds \) \(=\) \(\ds \paren {\sin \frac {3 \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {3 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {3 - 1} \theta} + \paren {\sin \frac {2 \pi} 2} \map \sin {\paren {3 - 2} \theta} + \paren {\sin \frac {3 \pi} 2} \map \sin {\paren {3 - 3} \theta} }\)

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:

$\ds \map \sin {n \theta} = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$

from which we are to show:

$\ds \map \sin {\paren {n + 2} \theta} = \paren {\sin \frac {\paren {n + 2} \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n + 2} \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n + 2 - k} \theta} }$


Induction Step

This is our induction step:

For $n$ even or for $n$ odd (the argument is identical):

\(\ds \map \sin {\paren {n + 2} \theta}\) \(=\) \(\ds \paren {2 \cos \theta} \map \sin {\paren {n + 1} \theta} - \map \sin {\paren n \theta}\) Sine of Integer Multiple of Argument: Formulation 4
\(\ds \) \(=\) \(\ds \paren {2 \cos \theta} \map \sin {\paren {n + 1} \theta} - \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} - 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\)
\(\ds \) \(=\) \(\ds - \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {n + 2 - 1} \theta} } - 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\) rearranging terms and adding zero
\(\ds \) \(=\) \(\ds \paren {\map \sin {\frac {n \pi} 2 + \pi} } \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {n + 2 - 1} \theta} } - 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }\) Sine of Angle plus Integer Multiple of Pi
\(\ds \) \(=\) \(\ds \paren {\sin \frac {\paren {n + 2} \pi} 2} \paren {\sin \theta} + \paren {2 \cos \theta} \paren {\paren {\sin \frac {0 \pi} 2} \map \sin {\paren {n + 2 - 0} \theta} + \paren {\sin \frac \pi 2} \map \sin {\paren {n + 2 - 1} \theta} } + 2 \cos \theta \paren {\sum_{k \mathop = 2}^{n + 2} \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n + 2 - k} \theta} }\) Translation of Index Variable of Summation
\(\ds \) \(=\) \(\ds \paren {\sin \frac {\paren {n + 2} \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^{n + 2} \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n + 2 - k} \theta} }\)


The result follows by the Principle of Mathematical Induction.

Therefore:

$\ds \forall n \in \Z_{>0}: \sin n \theta = \paren {\sin \frac {n \pi} 2} \paren {\sin \theta} + 2 \cos \theta \paren {\sum_{k \mathop = 0}^n \paren {\sin \frac {k \pi} 2} \map \sin {\paren {n - k} \theta} }$

$\blacksquare$


Examples

Sine of Quintuple Angle

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


Sine of Sextuple Angle

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