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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 Morrie's law ： ウィキペディア英語版 Morrie's law Morrie's law is a name that occasionally is used for the trigonometric identity :$\backslash cos(20^\backslash circ)\; \backslash cdot\; \backslash cos(40^\backslash circ)\; \backslash cdot\; \backslash cos(80^\backslash circ)=\backslash frac.$ It is a special case of the more general identity :$2^n\; \backslash cdot\; \backslash prod\_^\; \backslash cos(2^k\; \backslash alpha)=\backslash frac$ with ''n'' = 3 and α = 20° and the fact that :$\backslash frac\; =\; \backslash frac\; =\; 1$ since :$\backslash sin(180^\backslash circ-x)\; =\; \backslash sin(x).$ The name is due to the physicist Richard Feynman, who used to refer to the identity under that name. Feynman picked that name because he learned it during his childhood from a boy with the name Morrie Jacobs and afterwards remembered it for all of his life.〔W.A. Beyer, J.D. Louck, and D. Zeilberger, ''A Generalization of a Curiosity that Feynman Remembered All His Life'', Math. Mag. 69, 43–44, 1996.〕 A similar identity for the sine function also holds: :$\backslash sin(20^\backslash circ)\; \backslash cdot\; \backslash sin(40^\backslash circ)\; \backslash cdot\; \backslash sin(80^\backslash circ)=\backslash frac.$ Moreover, dividing the second identity by the first, the following identity is evident: :$\backslash tan(20^\backslash circ)\; \backslash cdot\; \backslash tan(40^\backslash circ)\; \backslash cdot\; \backslash tan(80^\backslash circ)=\backslash sqrt\; 3\; =\; \backslash tan(60^\backslash circ).\; \backslash ,$ ==Proof== Recall the double angle formula for the sine function :$\backslash sin(2\; \backslash alpha)\; =\; 2\; \backslash sin(\backslash alpha)\; \backslash cos(\backslash alpha).\; \backslash ,$ Solve for $\backslash cos(\backslash alpha)$ :$\backslash cos(\backslash alpha)=\backslash frac.$ It follows that: :$$ \begin \cos(2 \alpha) & = \frac \\() \cos(4 \alpha) & = \frac \\ & \alpha)}. \end Multiplying all of these expressions together yields: :$\backslash cos(\backslash alpha)\; \backslash cos(2\; \backslash alpha)\; \backslash cos(4\; \backslash alpha)\; \backslash cdots\; \backslash cos(2^\; \backslash alpha)=$ \frac \cdot \frac \cdot \frac \cdots \frac. The intermediate numerators and denominators cancel leaving only the first denominator, a power of 2 and the final numerator. Note that there are ''n'' terms in both sides of the expression. Thus, :$\backslash prod\_^\; \backslash cos(2^k\; \backslash alpha)=\backslash frac,$ which is equivalent to the generalization of Morrie's law. 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「Morrie's law」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.