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

\(\ds \cos z\) \(=\) \(\ds \cos \left({\dfrac {\pi} 2 + i \ln \phi}\right)\)
\(\ds \) \(=\) \(\ds \frac {e^{i \left({\left({\pi / 2}\right) + i \ln \phi}\right)} + e^{-i \left({\left({\pi / 2}\right) + i \ln \phi}\right)} } 2\) Cosine Exponential Formulation
\(\ds \) \(=\) \(\ds \frac {e^{i \pi / 2} e^{-\ln \phi} + e^{-i \pi / 2} e^{\ln \phi} } 2\)
\(\ds \) \(=\) \(\ds \frac {e^{-\ln \phi} \left({\cos \frac \pi 2 + i \sin \frac \pi 2}\right) + e^{\ln \phi} \left({\cos \left({-\frac \pi 2}\right) + i \sin \left({-\frac \pi 2}\right)}\right)} 2\) Euler's Formula and Corollary
\(\ds \) \(=\) \(\ds \frac {e^{-\ln \phi} \left({i \sin \frac \pi 2}\right) + e^{\ln \phi} \left({i \sin \left({-\frac \pi 2}\right)}\right)} 2\) 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:

\(\ds \sin \left({n + 1}\right) z + \sin \left({n - 1}\right) z\) \(=\) \(\ds 2 \sin n z \cos z\) Simpson's Formula for Sine by Cosine
\(\ds \) \(=\) \(\ds -i \sin n z\)


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Sources

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