Sine of Integer Multiple of Argument/Formulation 9
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Theorem
For $n \in \Z_{>1}$:
- $\sin n \theta = \map \cos {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
- $r = \begin {cases} 2 & : \text {$n$ is even} \\ 1 & : \text {$n$ is odd} \end {cases}$
- $a_k = \begin {cases} 2 \sin \theta & : \text {$k$ is even} \\ -2 \sin \theta & : \text {$k$ is odd and $k < n - 1$} \\ \sin \theta & : k = n - 1 \end {cases}$
Proof
| \(\ds \map \sin {n \theta}\) | \(=\) | \(\ds \paren {2 \sin \theta } \map \cos {\paren {n - 1 } \theta} + \map \sin {\paren {n - 2 } \theta}\) | Line 1: Sine of Integer Multiple of Argument/Formulation 6 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\paren {n - 1 } \theta} \paren { \paren {2 \sin \theta } + \frac {\map \sin {\paren {n - 2 } \theta} } {\map \cos {\paren {n - 1 } \theta} } }\) | Line 2: Factor out $\map \cos {\paren {n - 1 } \theta}$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {\cfrac {\map \cos {\paren {n - 1 } \theta} } {\map \sin {\paren {n - 2 } \theta} } } }\) | Line 3: Move the numerator to the denominator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {\cfrac {\paren {-2 \sin \theta } \map \sin {\paren {n - 2 } \theta} + \map \cos {\paren {n - 3 } \theta} } {\map \sin {\paren {n - 2 } \theta} } } }\) | Line 4: Cosine of Integer Multiple of Argument/Formulation 6 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac {\map \cos {\paren {n - 3 } \theta} } {\map \sin {\paren {n - 2 } \theta} } } }\) | Line 5: Simplify expression | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {\cfrac {\map \sin {\paren {n - 2 } \theta} } {\map \cos {\paren {n - 3 } \theta} } } } }\) | Line 6: Move the numerator to the denominator | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {\cfrac {\paren {2 \sin \theta } \map \cos {\paren {n - 3 } \theta} + \map \sin {\paren {n - 4 } \theta} } {\map \cos {\paren {n - 3} \theta} } } } }\) | Line 7: Sine of Integer Multiple of Argument/Formulation 6 | |||||||||||
| \(\ds \) | \(=\) | \(\ds \map \cos {\paren {n - 1 } \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {2 \sin \theta + \cfrac {\map \sin {\paren {n - 4 } \theta} } {\map \cos {\paren {n - 3} \theta} } } } }\) | Line 8: Simplify expression |
By comparing Line 2 to Line 8, we see that:
 | This article contains statements that are justified by handwavery. In particular: All I see on line 2 is an expression concerning $\map \cos {\paren {n - 1} \theta}$, and all I see on line 8 is another more complicated expression concerning $\map \cos {\paren {n - 1} \theta}$. Nothing about $\dfrac {\map \sin {\paren {n - \paren {2k } } \theta} } {\map \cos {\paren {n - \paren {2 k - 1} } \theta} }$. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding precise reasons why such statements hold. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Handwaving}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
| \(\ds \frac {\map \sin {\paren {n - \paren {2 k} } \theta} } {\map \cos {\paren {n - \paren {2 k - 1 } } \theta} }\) | \(=\) | \(\ds \paren {\cfrac 1 {-2 \sin \theta + \cfrac 1 {2 \sin \theta + \cfrac {\map \sin {\paren {n - 2 \paren {k + 1 } } \theta} } {\map \cos {\paren {n - \paren {2 \paren {k + 1} - 1} } \theta} } } } }\) |
Therefore, the terminal denominator will be:
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta} + \frac {\map \sin {\paren {n - \paren {2k } } \theta} } {\map \cos {\paren {n - \paren {2k - 1 } } \theta} }\) |
Assume $n$ even $n = 2k$
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta} + \frac {\map \sin {\paren {2k - \paren {2k } } \theta} } {\map \cos {\paren {2k - \paren {2k - 1 } } \theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta} + \frac {\map \sin {0} } {\map \cos {\theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta}\) |
Assume $n$ odd $n = 2k - 1$
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta} + \frac {\map \sin {\paren {2k - 1 - \paren {2k } } \theta} } {\map \cos {\paren {2k - 1 - \paren {2k - 1 } } \theta} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta} + \frac {\map \sin {-\theta} } {\map \cos 0}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {2 \sin \theta} - \frac {\map \sin {\theta} } {\map \cos 0}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {\sin \theta}\) |
Therefore:
- $\sin n \theta = \map \cos {\paren {n - 1} \theta} \paren {a_0 + \cfrac 1 {a_1 + \cfrac 1 {a_2 + \cfrac 1 {\ddots \cfrac {} {a_{n - r} } } } } }$
where:
- $r = \begin {cases} 2 & : \text {$n$ is even} \\ 1 & : \text {$n$ is odd} \end {cases}$
- $a_k = \begin {cases} 2 \sin \theta & : \text {$k$ is even} \\ -2 \sin \theta & : \text {$k$ is odd and $k < n - 1$} \\ \sin \theta & : k = n - 1 \end {cases}$
$\blacksquare$
Examples
Sine of Quintuple Angle
- $\map \sin {5 \theta } = \map \cos {4 \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {2\sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {\sin \theta }} }} }$
Sine of Sextuple Angle
- $\map \sin {6 \theta } = \map \cos {5 \theta} \paren { 2 \sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {2\sin \theta + \cfrac 1 {-2 \sin \theta + \cfrac 1 {2\sin \theta }} }} }$