# 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.

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

$\cos\,(6^{6^{6}})^{\circ}=-\phi/2.$

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 $\alpha +\beta =60^{\circ},$

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

Also

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

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.)

### Trigonometry

- What Is Trigonometry?
- Addition and Subtraction Formulas for Sine and Cosine
- The Law of Cosines (Cosine Rule)
- Cosine of 36 degrees
- Tangent of 22.5
^{o}- Proof Wthout Words - Sine and Cosine of 15 Degrees Angle
- Sine, Cosine, and Ptolemy's Theorem
- arctan(1) + arctan(2) + arctan(3) = π
- Trigonometry by Watching
- arctan(1/2) + arctan(1/3) = arctan(1)
- Morley's Miracle
- Napoleon's Theorem
- A Trigonometric Solution to a Difficult Sangaku Problem
- Trigonometric Form of Complex Numbers
- Derivatives of Sine and Cosine
- ΔABC is right iff sin²A + sin²B + sin²C = 2
- Advanced Identities
- Hunting Right Angles
- Point on Bisector in Right Angle
- Trigonometric Identities with Arctangents
- The Concurrency of the Altitudes in a Triangle - Trigonometric Proof
- Butterfly Trigonometry
- Binet's Formula with Cosines
- Another Face and Proof of a Trigonometric Identity
- cos/sin inequality
- On the Intersection of kx and |sin(x)|
- Cevians And Semicircles
- Double and Half Angle Formulas
- A Nice Trig Formula
- Another Golden Ratio in Semicircle
- Leo Giugiuc's Trigonometric Lemma
- Another Property of Points on Incircle
- Much from Little
- The Law of Cosines and the Law of Sines Are Equivalent
- Wonderful Trigonometry In Equilateral Triangle
- A Trigonometric Observation in Right Triangle
- A Quick Proof of cos(pi/7)cos(2.pi/7)cos(3.pi/7)=1/8

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