Sine of Multiple of Pi by 2 plus i by Natural Logarithm of Golden Mean/Proof 2
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Theorem
Let $z = \dfrac \pi 2 + i \ln \phi$.
Then:
- $\dfrac {\sin n z} {\sin z} = i^{1 - n} F_n$
where:
- $\phi$ denotes the golden mean
- $F_n$ denotes the $n$th Fibonacci number.
Proof
\(\ds \cos z\) | \(=\) | \(\ds \map \cos {\dfrac \pi 2 + i \ln \phi}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{i \paren {\paren {\pi / 2} + i \ln \phi} } + e^{-i \paren {\paren {\pi / 2} + i \ln \phi} } } 2\) | Euler's Cosine Identity | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{i \pi / 2} e^{-\ln \phi} + e^{-i \pi / 2} e^{\ln \phi} } 2\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{-\ln \phi} \paren {\cos \frac \pi 2 + i \sin \frac \pi 2} + e^{\ln \phi} \paren {\map \cos {-\frac \pi 2} + i \map \sin {-\frac \pi 2} } } 2\) | Euler's Formula and {Corollary |
}} | ||||||||||
\(\ds \) | \(=\) | \(\ds \frac {e^{-\ln \phi} \paren {i \sin \frac \pi 2} + e^{\ln \phi} \paren {i \map \sin {-\frac \pi 2} } } 2\) | Cosine of Half-Integer Multiple of Pi | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {i e^{-\ln \phi} - i e^{\ln \phi} } 2\) | Sine of Half-Integer Multiple of Pi and simplification | |||||||||||
\(\ds \) | \(=\) | \(\ds -i \frac {\phi - \frac 1 \phi} 2\) | Exponential of Natural Logarithm | |||||||||||
\(\ds \) | \(=\) | \(\ds -i \frac {\phi^2 - 1} {2 \phi}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -i \frac \phi {2 \phi}\) | Square of Golden Mean equals One plus Golden Mean | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {-i} 2\) |
Then:
\(\ds \map \sin {n + 1} z + \map \sin {n - 1} z\) | \(=\) | \(\ds 2 \sin n z \cos z\) | Werner Formula for Sine by Cosine | |||||||||||
\(\ds \) | \(=\) | \(\ds -i \sin n z\) |
This needs considerable tedious hard slog to complete it. In particular: Where to from here? 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 {{Finish}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1997: Donald E. Knuth: The Art of Computer Programming: Volume 1: Fundamental Algorithms (3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $33$: Solution