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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 Skinny triangle ： ウィキペディア英語版 Skinny triangle A skinny triangle in trigonometry is a triangle whose height is much greater than its base. The solution of such triangles can be greatly simplified by using the approximation that the sine of a small angle is equal to the angle in radians. The solution is particularly simple for skinny triangles that are also isosceles or right triangles: in these cases the need for trigonometric functions or tables can be entirely dispensed with. The skinny triangle finds uses in surveying, astronomy and shooting. ==Isosceles triangle== |} The approximated solution to the skinny isosceles triangle, referring to figure 1, is; :$b\; \backslash simeq\; r\; \backslash theta\; \backslash ,$ :$area\; \backslash simeq\; \backslash frac12\; \backslash theta\; r^2\; \backslash ,$ This is based on the small-angle approximations; :$\backslash sin\; \backslash theta\; \backslash simeq\; \backslash theta,\; \backslash quad\; \backslash theta\; \backslash ll\; 1\; \backslash ,$ and, :$\backslash cos\; \backslash theta\; =\; \backslash sin\; \backslash left\; (\; \backslash frac-\backslash theta\; \backslash right\; )\; \backslash simeq\; 1,\; \backslash quad\; \backslash theta\; \backslash ll\; 1$ when $\backslash scriptstyle\; \backslash theta$ is in radians. The proof of the skinny triangle solution follows from the small-angle approximation by applying the law of sines. Again referring to figure 1; :$\backslash frac\; =\; \backslash frac\; \backslash right\; )\}$ The term $\backslash scriptstyle\; \backslash frac$ represents the base angle of the triangle and is this value because the sum of the internal angles of any triangle (in this case the two base angles plus ''θ'') are equal to ''π''. Applying the small angle approximations to the law of sines above results in; :$\backslash frac\; \backslash simeq\; \backslash frac$ the desired result. This result is equivalent to assuming that the length of the base of the triangle is equal to the length of the arc of circle of radius ''r'' subtended by angle ''θ''. This approximation becomes ever more accurate for smaller and smaller ''θ''. The error is 10% or less for angles less than about 43°.〔Abell ''et al.'', pp. 414–415〕〔Breithaupt, p. 26〕 The side-angle-side formula for the area of the triangle is; :$area\; =\; \backslash frac2\; r^2$ Applying the small angle approximations results in; :$area\; \backslash simeq\; \backslash frac12\; \backslash theta\; r^2\; \backslash ,$ 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「Skinny triangle」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.