Value of Golden Ratio using 666

Theorem

The Golden Ratio can be given by the following formula:

$\phi = -2 \sin 666 \degrees = 1.61803 \, 39887 \, 5 \ldots$


Proof

\(\ds -2 \sin 666 \degrees\) \(=\) \(\ds -2 \sin 306 \degrees\) Sine of Angle plus Full Angle
\(\ds \) \(=\) \(\ds 2 \sin 54 \degrees\) Sine of Conjugate Angle
\(\ds \) \(=\) \(\ds 2 \cos 36 \degrees\) Sine of Complement equals Cosine
\(\ds \) \(=\) \(\ds 2 \times \frac \phi 2\) Cosine of $36 \degrees$
\(\ds \) \(=\) \(\ds \phi\)

$\blacksquare$


Sources

  • 1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $666$
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