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Through Taylor series, is it possible to say that the cos(x) or sin(x) of any value other than the nice ones {(pi/3), (pi/6), and finally the ((z*pi)/2)} are non commensurable with a 1 unit measure? If yes, prove (although that should go without saying).

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    So far, my intuition have been that since we have an infinite amount of summation terms, and each of these terms has a different denominator than the other, our number would necessarily be irrational. Thus it is non-commensurable the sin(x) and cos(x) values would be non-commensurable with a 1 unit entity, no matter how small it is made to be. – 2017-01-30
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    I would be surprised to see how power series would be useful for this. The ue of Eisentein's Criterion and de Moivre's Theorem will solve this. – 2017-01-30
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    @PaulD.P.Blum: *"[M]y intuition have been that since we have an infinite amount of summation terms, and each of these terms has a different denominator than the other, our number would necessarily be irrational"* ... What about $\frac12 + \frac14 + \frac18 + \frac1{16} + \cdots$? Infinitely many terms, each with a different denominator, contributing to a decidedly non-irrational value. – 2017-02-01
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    to user254665, those two theories are more complicated than are the Taylor / Power series. Perhaps my statement is true, I am just an amature mathematician, and primarily a philosopher, after all. – 2017-02-07

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A Google search for "irrationality of cosine" turns up a number of relevant links, including these:

http://people.math.sc.edu/filaseta/gradcourses/Math785/Math785Notes2.pdf

This one proves that The number π is irrational and For any rational number α $\ne$ 0, cos α is irrational.

https://en.wikipedia.org/wiki/Niven's_theorem

This says:

In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0 ≤ θ ≤ 90 for which the sine of θ degrees is also a rational number are:[1]

${\displaystyle {\begin{aligned}\sin 0^{\circ }&=0,\\[10pt]\sin 30^{\circ }&={\frac {1}{2}},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}} {\begin{aligned}\sin 0^{\circ }&=0,\\[10pt]\sin 30^{\circ }&={\frac {1}{2}},\\[10pt]\sin 90^{\circ }&=1.\end{aligned}}$

In radians, one would require that 0 ≤ x ≤ π/2, that x/π be rational, and that sin x be rational. The conclusion is then that the only such values are sin 0 = 0, sin π/6 = 1/2, and sin π/2 = 1.

The theorem appears as Corollary 3.12 in Niven's book on irrational numbers.[2]

The theorem extends to the other trigonometric functions as well.[2] For rational values of θ, the only rational values of the sine or cosine are 0, ±1/2, and ±1; the only rational values of the secant or cosecant are ±1 and ±2; and the only rational values of the tangent or cotangent are 0 and ±1.

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    +1 for mentioning Niven's book *Irrational Numbers*. It's truly a gem and can be studied by anyone with a basic knowledge of calculus. – 2017-01-31