A silly proof of the fact that the golden ratio is irrational by identifying it as the cosine of a rational angle

Since the golden ratio has popped up twice now more or less by accident I would like to add my own silly proof of the fact that the golden ratio is irrational. Okay, it’s not my proof so much as it is a combination of two separate results which I like.

So just a recap of the concept. The golden ratio is a particular number \varphi which to my knowledge was first remarked on in Euclids Elements where it was defined as a particular proportion a/b which could was also equal to (a + b)/a which could be read as “a + b is to a as a is to b”..

\varphi = \cfrac{a}{b} + \cfrac{a + b}{a}

Algebraically this gives rise to the equation \varphi^2 - \varphi - 1 = 0 which you can solve for

\varphi = \cfrac{1}{2} (1 + \sqrt{5}) = 1.6180033987...

The fact that this number is irrational is obvious as it is the sum of an irrational number and a rational number and you could also go straight from the definition and find that a and b cannot be integers.

This makes the issue of finding yet another way of proving it to be irrational seem rather redundant but it ties into the fact that you can actually identify the golden ratio as belonging to an absolutely massive class of irrational numbers which might not be well known. The cosines of rational angles.

You see we have the curious fact that

\varphi = 2\cos(\pi/5) = 2\cos 36^\circ

which may been seen to come from the golden ratio being the proportion of the the longer and the shorter side of an isosceles triangle with angles 72^\circ, \; 72^\circ, \; 36^\circ; the so called golden triangle, (Or we may recover it from trigonometry or viewing the equation of the golden ratio as the real part of a particular trigonometric identity)

Now it actually is the case that almost all cosines of rational angles are actually irrational numbers, all but 3 (and their trigonometric conjugates) to be precise. In case the term rational angle is not familiar it means precisely what it sounds like if we work in degrees instead of radians. 22^\circ is a rational angle, 0.5^\circ is a rational angle and in general  (p/q)^\circ is a rational angle. In radians it is rather taken to mean that the angle is rational with respect to \pi so an angle in radians is rational if it can be written as (p/q) \pi.

The only angles which have rational cosines are those such that \cos \alpha \in \{-1,-1/2, 0, 1/2, 1\} or if we spell the anglesout explicitely in degrees in the range 0^\circ to 360^\circ we get

0^\circ, \;, 60^\circ, \; 90^\circ, \; 120^\circ, 180^\circ, \;140^\circ, 270^\circ, \; 300^\circ, \; 360^\circ

The golden ratio with its 36^\circ obviously doesn’t fit into this so it is irrational!

A simple proof of the fact that these are the only rational angles with rational cosines can be found in this arxiv article by Jord Jahnel: When is the (co)sine of a rational angle equal to a rational number? along with some other related results.

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