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 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 of $0, 1, -1$ 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.^[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 $\tan \theta.$

References

1 2 Niven, Ivan. Numbers: Rational and Irrational, 1961.
1 2 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. http://arxiv.org/abs/0911.1933
↑ Weisstein, Eric W. "Trigonometry Angles." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TrigonometryAngles.html

Irrational numbers

Prime ( $ρ$ ) Euler–Mascheroni ( $γ$ ) Logarithm of 2 Twelfth root of two (¹²√2) Apéry's ( $ζ (3)$ ) Sophomore's dream Plastic ( $ρ$ ) Square root of 2 (√2) Erdős–Borwein ( $E$ ) Golden ratio ( $φ$ ) Square root of 3 (√3) Square root of 5 (√5) Silver ratio ( $δ S$ ) Second Feigenbaum ( $α$ ) Euler's ( $e$ ) Pi ( $π$ ) First Feigenbaum ( $δ$ )

Schizophrenic Transcendental Trigonometric

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Trigonometric number

See also

References