Value of Golden Ratio using 666
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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): $666$