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.

A real number different from 0, 1, –1 is a trigonometric number if and only if it is the real part of a root of unity.

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. Thus every trigonometric number is an algebraic number. This latter statement can be proved[2]:pp. 29-30 by starting 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 ${\displaystyle \pi ,}$ is algebraic as can 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 ${\displaystyle \tan \theta .}$