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 circle, or equivalently, the sine or cosine in radians of a rational multiple of π, or the sine or cosine of a rational number of degrees.

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.^[4] For example,

$\cos (\pi / 23)=-(1/2)(-1)^{22/23}(1+(-1)^{2/23}).$

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 $\theta = 2\pi k/n$ for coprime k and n:

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

Expanding the left side and equating real parts gives an equation in $\cos \theta$ and $\sin^2 \theta;$ substituting $\sin^2 \theta =1-\cos^2 \theta$ gives a polynomial equation having $\cos \theta$ as a solution, so by definition the latter is an algebraic number. Also $\sin \theta$ is algebraic since it equals the algebraic number $\cos(\theta-\pi /2).$ Finally, $\tan \theta,$ where again $\theta$ is a rational multiple of $\pi,$ is algebraic as can be seen by equating the imaginary parts of the expansion of the de Moivre equation and dividing through by $\cos^n \theta$ to obtain a polynomial equation in $\tan \theta.$

See also

• Exact trigonometric constants
• Niven's theorem

References

1. Niven, Ivan. Numbers: Rational and Irrational, 1961.
2. Niven, Ivan. Irrational Numbers, Carus Mathematical Monographs no. 11, 1956.
3. Li Zhou and Lubomir Markov (2010). "Recurrent Proofs of the Irrationality of Certain Trigonometric Values". American Mathematical Monthly 117 (4): 360–362. doi:10.4169/000298910x480838. http://arxiv.org/abs/0911.1933
4. Weisstein, Eric W. "Trigonometry Angles." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TrigonometryAngles.html