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
\(\ds \) \(=\) \(\ds -i \sin n z\)




Sources