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

References

  1. 1 2 Niven, Ivan. Numbers: Rational and Irrational, 1961.
  2. 1 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): 360362. 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
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