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 for coprime k and n:
Expanding the left side and equating real parts gives an equation in and substituting gives a polynomial equation having as a solution, so by definition the latter is an algebraic number. Also is algebraic since it equals the algebraic number Finally, where again is a rational multiple of 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 to obtain a polynomial equation in
See also
References
- ^ ^{a} ^{b} Niven, Ivan. Numbers: Rational and Irrational, 1961.
- ^ ^{a} ^{b} Niven, Ivan. Irrational Numbers, Carus Mathematical Monographs no. 11, 1956.
- ^ 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. https://arxiv.org/abs/0911.1933