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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Trigonometric number ： ウィキペディア英語版 Trigonometric number In mathematics, a trigonometric number〔 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.〔Niven, Ivan. ''Numbers: Rational and Irrational'', 1961.〕〔Niven, Ivan. ''Irrational Numbers'', Carus Mathematical Monographs no. 11, 1956.〕 Li Zhou and Lubomir Markov〔 http://arxiv.org/abs/0911.1933〕 recently improved and simplified Niven's proofs. Any trigonometric number can be expressed in terms of radicals.〔Weisstein, Eric W. "Trigonometry Angles." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/TrigonometryAngles.html〕 For example, :$\backslash cos\; (\backslash pi\; /\; 23)=-(1/2)(-1)^(1+(-1)^).$ Thus every trigonometric number is an algebraic number. This latter statement can be proved〔 by starting with the statement of de Moivre's formula for the case of $\backslash theta\; =\; 2\backslash pi\; k/n$ for coprime ''k'' and ''n'': :$(\backslash cos\; \backslash theta\; +\; i\; \backslash sin\; \backslash theta\; )^n\; =1.$ Expanding the left side and equating real parts gives an equation in $\backslash cos\; \backslash theta$ and $\backslash sin^2\; \backslash theta;$ substituting $\backslash sin^2\; \backslash theta\; =1-\backslash cos^2\; \backslash theta$ gives a polynomial equation having $\backslash cos\; \backslash theta$ as a solution, so by definition the latter is an algebraic number. Also $\backslash sin\; \backslash theta$ is algebraic since it equals the algebraic number $\backslash cos(\backslash theta-\backslash pi\; /2).$ Finally, $\backslash tan\; \backslash theta,$ where again $\backslash theta$ is a rational multiple of $\backslash pi,$ is algebraic as can be seen by equating the imaginary parts of the expansion of the de Moivre equation and dividing through by $\backslash cos^n\; \backslash theta$ to obtain a polynomial equation in $\backslash tan\; \backslash theta.$ == See also == * Exact trigonometric constants * Niven's theorem 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Trigonometric number」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.