Beauty and the Beast - in Trigonometry

The term the Number of the Beast commonly refers to $666,$ with the origin in the Book of Revelations, of the New Testament. Here is a painting by William Blake to heighten the association.

Number of the beast by William Blake

The Golden Ratio $\phi,$ often referred to as the Divine Proportion, has long been associated with beauty in natural and artificial forms.

Starting with a known fact that $\cos 36^{\circ}=\phi /2,$ Daniel J. Hardisky has found multiple relations involving both the Number of the Beast and the Golden Ratio - in trigonometry.

For example, $\sin 666^{\circ}= -\phi /2.$ Indeed,

$\begin{align} \sin 666^{\circ}&=\sin 306^{\circ}\\ &=-\sin 54^{\circ}\\ &=-\cos 36^{\circ} = -\phi /2, \end{align}$

where we used a couple of basic trigonometric formulas, $\sin (90^{\circ}-x)=\cos x$ and $\sin (360^{\circ}+x)=\sin x.$ Further,

$\begin{align} \cos\,(6\times 6\times 6)^{\circ}&=\cos 216^{\circ}\\ &=\cos (-144^{\circ})\\ &=-\cos 36^{\circ} = -\phi /2, \end{align}$

giving together $\sin 666^{\circ}+\cos\,(6\times 6\times 6)^{\circ}=-\phi.$ And, since $\displaystyle 6^{6^{6}}\equiv 216\mod 360,$ it is also true that


Next come $\cos 6^{\circ}-\cos 66^{\circ}=\cos 666^{\circ}$ and $\sin 6^{\circ}-\sin 66^{\circ}=\sin 666^{\circ}$ that show that the Number of the Beast comes in an extended family. To prove the first use that, for $\alpha +\beta =120^{\circ},$

$\cos\alpha +\cos\beta=\cos (60^{\circ}-\alpha )=\cos (60^{\circ}-\beta ).$

For the second, prove that, in general, for $\alpha +\beta =60^{\circ},$

$\sin\alpha +\sin\beta=\cos (60^{\circ}+\alpha )=\sin (60^{\circ}+\beta ).$


$\tan 666^{\circ}\cdot\tan \,(6\times 6\times 6)^{\circ}=-1.$

There is an infinite sequence of identities following from $666^{666}\equiv 216\mod 360$ and $666^{216}\equiv 216\mod 360$ so that, modulo $360,$

$\displaystyle \cos (666^{666})^{\circ}\equiv \cos (666^{666^{666}})^{\circ}\equiv \cos (666^{666^{666^{666}}})^{\circ}\ldots =-\phi /2.$

(Daniel notes that expressions for $\sin 666^{\circ}$ and $\cos (6\cdot 6\cdot 6)^{\circ}$ came from an article by Steve C. Wang, Journal of Recreational Mathematics, Vol. 26, Number 3.)

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