Sine of Multiple of Pi by 2 plus i by Natural Logarithm of Golden Mean/Proof 1
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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
| \(\ds \sin n z\) | \(=\) | \(\ds \map \sin {\dfrac {n \pi} 2 + i n \ln \phi}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{i \paren {\paren {n \pi / 2} + i n \ln \phi} } - e^{-i \paren {\paren {n \pi / 2} + i n \ln \phi} } } {2 i}\) | Sine Exponential Formulation | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{i n \pi / 2} e^{-n \ln \phi} - e^{-i n \pi / 2} e^{n \ln \phi} } {2 i}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{-n \ln \phi} \paren {\cos \frac {n \pi} 2 + i \sin \frac {n \pi} 2} - e^{n \ln \phi} \paren {\cos \paren {-\frac {n \pi} 2} + i \map \sin {-\frac {n \pi} 2} } } {2 i}\) | Euler's Formula and Corollary | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {e^{-n \ln \phi} \paren {i \sin \frac {n \pi} 2} - e^{n \ln \phi} \paren {i \map \sin {-\frac {n \pi} 2} } } {2 i}\) | Cosine of Half-Integer Multiple of Pi | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {i^n e^{-n \ln \phi} - \paren {-i}^n e^{n \ln \phi} } {2 i}\) | Sine of Half-Integer Multiple of Pi and simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac {i^{n - 1} \paren {e^{-n \ln \phi} + e^{n \ln \phi} } } 2\) | simplification | |||||||||||
| \(\ds \) | \(=\) | \(\ds i^{n - 1} \frac {\phi^n + \frac 1 {\phi^n} } 2\) | Exponential of Natural Logarithm | |||||||||||
| \(\ds \) | \(=\) | \(\ds i^{n - 1} \frac {\phi^n - \paren {-\frac 1 {\phi^n} } } 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i^{n - 1} \frac {\phi^n - \hat \phi^n} 2\) | Reciprocal Form of One Minus Golden Mean |
Setting $n = 1$:
- $\sin z = i^0 \frac {\phi^1 - \hat \phi^1} 2 = \frac {\phi - \hat \phi} 2$
Thus:
| \(\ds \dfrac {\sin n z} {\sin z}\) | \(=\) | \(\ds \dfrac {i^{n - 1} \frac {\phi^n - \hat \phi^n} 2} {\frac {\phi - \hat \phi} 2}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i^{n - 1} \dfrac {\phi^n - \hat \phi^n } {\phi - \hat \phi}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i^{n - 1} \dfrac {\frac {\phi^n - \hat \phi^n} {\sqrt 5} } {\frac {\phi - \hat \phi} {\sqrt 5} }\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds i^{n - 1} \dfrac {F_n} {F_1}\) | Euler-Binet Formula | |||||||||||
| \(\ds \) | \(=\) | \(\ds i^{n - 1} F_n\) | Definition of Fibonacci Number: $F_1 = 1$ |
$\blacksquare$