Trigonometric number

In mathematics, a trigonometric number[1]^{:ch. 5} is an irrational number produced by taking the sine or cosine of a rational multiple of a full circle, or equivalently, the sine or cosine of an angle which in radians is a rational multiple of π, or the sine or cosine of a rational number of degrees. One of the simplest examples is ${\displaystyle \cos {\frac {\pi }{4}}={\frac {\sqrt {2}}{2}}.}$

A real number different from 0, 1, –1, 1/2, –1/2 is a trigonometric number if and only if it is the real part of a root of unity (see Niven's theorem). Thus every trigonometric number is half the sum of two complex conjugate roots of unity. This implies that a trigonometric number is an algebraic number, and twice a trigonometric number is an algebraic integer.

Ivan Niven gave proofs of theorems regarding these numbers.[1][2]^{:ch. 3} Li Zhou and Lubomir Markov[3] recently improved and simplified Niven's proofs.

Any trigonometric number can be expressed in terms of radicals. Those that can be expressed in terms of square roots are well characterized (see Trigonometric constants expressed in real radicals). To express the other ones in terms of radicals, one requires nth roots of non-real complex numbers, with n > 2.

An elementary proof that every trigonometric number is an algebraic number is as follows.[2]^{:pp. 29–30}. One starts with the statement of de Moivre's formula for the case of ${\displaystyle \theta =2\pi k/n}$ for coprime k and n:

${\displaystyle (\cos \theta +i\sin \theta )^{n}=1.}$

Expanding the left side and equating real parts gives an equation in ${\displaystyle \cos \theta }$ and ${\displaystyle \sin ^{2}\theta$ ;} substituting ${\displaystyle \sin ^{2}\theta =1-\cos ^{2}\theta }$ gives a polynomial equation having ${\displaystyle \cos \theta }$ as a solution, so by definition the latter is an algebraic number. Also ${\displaystyle \sin \theta }$ is algebraic since it equals the algebraic number ${\displaystyle \cos(\theta -\pi /2).}$ Finally, ${\displaystyle \tan \theta ,}$ where again ${\displaystyle \theta }$ is a rational multiple of π, is algebraic as being the ratio of two algebraic numbers. In a more elementary way, this can also be seen by equating the imaginary parts of the two sides of the expansion of the de Moivre equation to each other and dividing through by ${\displaystyle \cos ^{n}\theta }$ to obtain a polynomial equation in ${\displaystyle \tan \theta .}$

References

1. Niven, Ivan. Numbers: Rational and Irrational, 1961. Random House. New Mathematical Library, Vol. 1. ISSN 0548-5932.
2. Niven, Ivan. Irrational Numbers, Carus Mathematical Monographs no. 11, 1956. Cambridge University Press (2005): ISBN 9780883850381.
3. Zhou, Li and Markov, Lubomir (2010). "Recurrent Proofs of the Irrationality of Certain Trigonometric Values". American Mathematical Monthly. 117 (4): 360–362. arXiv:0911.1933. doi:10.4169/000298910x480838. S2CID 19311924.CS1 maint: multiple names: authors list (link)