Difference between revisions of "Golden ratio"
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$$2\sin(i \log(\varphi))=i,$$ | $$2\sin(i \log(\varphi))=i,$$ | ||
where $\sin$ denotes the [[sine]] function, $i$ denotes the [[imaginary number]], $\log$ denotes the [[logarithm]], and $\varphi$ denotes the [[golden ratio]]. | where $\sin$ denotes the [[sine]] function, $i$ denotes the [[imaginary number]], $\log$ denotes the [[logarithm]], and $\varphi$ denotes the [[golden ratio]]. | ||
| + | <div class="mw-collapsible-content"> | ||
| + | <strong>Proof:</strong> █ | ||
| + | </div> | ||
| + | </div> | ||
| + | |||
| + | <div class="toccolours mw-collapsible mw-collapsed"> | ||
| + | <strong>Theorem:</strong> The following formula holds: | ||
| + | $$2\cos(i \log(1+\varphi))=3,$$ | ||
| + | where $\cos$ denotes the [[cosine]] function, $i$ denotes the [[imaginary number]], $\log$ denotes the [[logarithm]], and $\varphi$ denotes the [[golden ratio]]. | ||
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
Revision as of 04:52, 11 April 2015
The golden ratio is $\varphi = \dfrac{1+\sqrt{5}}{2}.$
Properties
Theorem: The following formula holds: $$2\sin(i \log(\varphi))=i,$$ where $\sin$ denotes the sine function, $i$ denotes the imaginary number, $\log$ denotes the logarithm, and $\varphi$ denotes the golden ratio.
Proof: █
Theorem: The following formula holds: $$2\cos(i \log(1+\varphi))=3,$$ where $\cos$ denotes the cosine function, $i$ denotes the imaginary number, $\log$ denotes the logarithm, and $\varphi$ denotes the golden ratio.
Proof: █
