Trigonometric number

In mathematics, a trigonometric number ^[1] is an irrational number obtained as the sine or cosine of a rational number of revolutions or, equivalently, the sine or cosine of an angle whose value in radians is a rational multiple of pi , or sine or cosine of the rational number of degrees .

A real number other than 0, 1, −1 is a trigonometric number if and only if it is the real part of the root of unity .

The proof of the theorems on these numbers was given by the Canadian-American mathematician Ivan Niven ^[1] , later his proofs were improved and simplified by Li Zhou and Lubomir Markov ^[2] .

Any trigonometric number can be expressed in terms of radicals . Thus, each trigonometric number is an algebraic number . The last assertion can be proved ^[1] , taking as a basis the Moire formula for the case $\theta =2\pi k/n$ ${\ displaystyle \ theta = 2 \ pi k / n}$ ${\ displaystyle \ theta = 2 \ pi k / n}$ for coprime k and n:

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

{\ displaystyle (\ cos \ theta + i \ sin \ theta) ^ {n} = 1.}

{\ displaystyle (\ cos \ theta + i \ sin \ theta) ^ {n} = 1.}

Extending the left side and equating the material parts gives the equation in $\cos \theta$ ${\ displaystyle \ cos \ theta}$ $\ cos \ theta$ and $\sin ^{2}\theta ;$ ${\ displaystyle \ sin ^ {2} \ theta;}$ ${\ displaystyle \ sin ^ {2} \ theta;}$ substituting $\sin ^{2}\theta =1-\cos ^{2}\theta$ ${\ displaystyle \ sin ^ {2} \ theta = 1- \ cos ^ {2} \ theta}$ ${\ displaystyle \ sin ^ {2} \ theta = 1- \ cos ^ {2} \ theta}$ , we obtain the polynomial equation having $\cos \theta$ ${\ displaystyle \ cos \ theta}$ $\ cos \ theta$ by its decision, therefore the latter is by definition an algebraic number. Also $\sin \theta$ ${\ displaystyle \ sin \ theta}$ $\ sin \ theta$ is an algebraic number because it is equal to an algebraic number $\cos(\theta -\pi /2).$ ${\ displaystyle \ cos (\ theta - \ pi / 2).}$ ${\ displaystyle \ cos (\ theta - \ pi / 2).}$ Finally, $\tan \theta$ ${\ displaystyle \ tan \ theta}$ ${\ displaystyle \ tan \ theta}$ where $\theta$ ${\ displaystyle \ theta}$ $\ theta$ is rational, multiple $\pi$ ${\ displaystyle \ pi}$ $\ pi$ is algebraic, which can be obtained by equating the imaginary parts of the two sides of the decomposition of the Muavre equation with each other and dividing by $\cos ^{n}\theta$ ${\ displaystyle \ cos ^ {n} \ theta}$ ${\ displaystyle \ cos ^ {n} \ theta}$ to obtain a polynomial equation in $\tan \theta .$ ${\ displaystyle \ tan \ theta.}$ ${\ displaystyle \ tan \ theta.}$

Notes

↑ ¹ ² ³ Niven, Ivan. Irrational Numbers , Carus Mathematical Monographs no. 11, 1956.
↑ Li Zhou and Lubomir Markov. Recurrent Proofs of the Irrationality of Certain Trigonometric Values (English) // American Mathematical Monthly : journal. - 2010 .-- Vol. 117 . - P. 360-362 . - DOI : 10.4169 / 000298910x480838 . https://arxiv.org/abs/0911.1933

[Niven-1] ¹ ² ³ Niven, Ivan. Irrational Numbers , Carus Mathematical Monographs no. 11, 1956.

[2] Li Zhou and Lubomir Markov. Recurrent Proofs of the Irrationality of Certain Trigonometric Values (English) // American Mathematical Monthly : journal. - 2010 .-- Vol. 117 . - P. 360-362 . - DOI : 10.4169 / 000298910x480838 . https://arxiv.org/abs/0911.1933

Trigonometric number

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