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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 Spherical law of cosines ： ウィキペディア英語版 Spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides〔) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points , and on the sphere (shown at right). If the lengths of these three sides are (from to (from to ), and (from to ), and the angle of the corner opposite is , then the (first) spherical law of cosines states:〔Romuald Ireneus 'Scibor-Marchocki, (Spherical trigonometry ), ''Elementary-Geometry Trigonometry'' web page (1997).〕〔W. Gellert, S. Gottwald, M. Hellwich, H. Kästner, and H. Küstner, ''The VNR Concise Encyclopedia of Mathematics'', 2nd ed., ch. 12 (Van Nostrand Reinhold: New York, 1989).〕 :$\backslash cos(c)\; =\; \backslash cos(a)\; \backslash cos(b)\; +\; \backslash sin(a)\; \backslash sin(b)\; \backslash cos(C).\; \backslash ,$ Since this is a unit sphere, the lengths , and are simply equal to the angles (in radians) subtended by those sides from the center of the sphere (for a non-unit sphere, they are the distances divided by the radius). As a special case, for , then , and one obtains the spherical analogue of the Pythagorean theorem: :$\backslash cos(c)\; =\; \backslash cos(a)\; \backslash cos(b).\; \backslash ,$ A variation on the law of cosines, the second spherical law of cosines, (also called the cosine rule for angles〔) states: :$\backslash cos(A)\; =\; -\backslash cos(B)\backslash cos(C)\; +\; \backslash sin(B)\backslash sin(C)\backslash cos(a)\; \backslash ,$ where and are the angles of the corners opposite to sides and , respectively. It can be obtained from consideration of a spherical triangle dual to the given one. If the law of cosines is used to solve for , the necessity of inverting the cosine magnifies rounding errors when is small. In this case, the alternative formulation of the law of haversines is preferable.〔R. W. Sinnott, "Virtues of the Haversine", Sky and Telescope 68 (2), 159 (1984).〕 ==Proof== A proof of the law of cosines can be constructed as follows.〔 Let , and denote the unit vectors from the center of the sphere to those corners of the triangle. Then, the lengths (angles) of the sides are given by the dot products: :$\backslash cos(a)\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$ :$\backslash cos(b)\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$ :$\backslash cos(c)\; =\; \backslash mathbf\; \backslash cdot\; \backslash mathbf$ To get the angle , we need the tangent vectors and at along the directions of sides and , respectively. For example, the tangent vector is the unit vector perpendicular to in the plane, whose direction is given by the component of perpendicular to . This means: :$\backslash mathbf\_a\; =\; \backslash frac\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)\}\; (\backslash mathbf\; \backslash cdot\; \backslash mathbf)\; \backslash right\backslash |\}\; =\; \backslash frac\; \backslash cos(a)\}$ where for the denominator we have used the Pythagorean identity and where || || denotes the length of the vector in the denominator. Similarly, :$\backslash mathbf\_b\; =\; \backslash frac\; \backslash cos(b)\}.$ Then, the angle is given by: :$\backslash cos(C)\; =\; \backslash mathbf\_a\; \backslash cdot\; \backslash mathbf\_b\; =\; \backslash frac$ from which the law of cosines immediately follows. 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「Spherical law of cosines」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.