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In trigonometry, the law of cosines (also known as the cosine formula or cosine rule) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using notation as in Fig. 1, the law of cosines states : where denotes the angle contained between sides of lengths ''a'' and ''b'' and opposite the side of length ''c''. The law of cosines generalizes the Pythagorean theorem, which holds only for right triangles: if the angle is a right angle (of measure 90° or π/2 radians), then and thus the law of cosines reduces to the Pythagorean theorem: : The law of cosines is useful for computing the third side of a triangle when two sides and their enclosed angle are known, and in computing the angles of a triangle if all three sides are known. By changing which sides of the triangle play the roles of ''a'', ''b'', and ''c'' in the original formula, the following two formulas also state the law of cosines: : : Though the notion of the cosine was not yet developed in his time, Euclid's ''Elements'', dating back to the 3rd century BC, contains an early geometric theorem almost equivalent to the law of cosines. The cases of obtuse triangles and acute triangles (corresponding to the two cases of negative or positive cosine) are treated separately, in Propositions 12 and 13 of Book 2. Trigonometric functions and algebra (in particular negative numbers) being absent in Euclid's time, the statement has a more geometric flavor: Using notation as in Fig. 2, Euclid's statement can be represented by the formula : This formula may be transformed into the law of cosines by noting that Proposition 13 contains an entirely analogous statement for acute triangles. The theorem was popularized in the Western world by François Viète in the 16th century. At the beginning of the 19th century, modern algebraic notation allowed the law of cosines to be written in its current symbolic form. ==Applications== The theorem is used in triangulation, for solving a triangle or circle, i.e., to find (see Figure 3): *the third side of a triangle if one knows two sides and the angle between them: :: *the angles of a triangle if one knows the three sides: :: *the third side of a triangle if one knows two sides and an angle opposite to one of them (one may also use the Pythagorean theorem to do this if it is a right triangle): :: These formulas produce high round-off errors in floating point calculations if the triangle is very acute, i.e., if ''c'' is small relative to ''a'' and ''b'' or ''γ'' is small compared to 1. It is even possible to obtain a result slightly greater than one for the cosine of an angle. The third formula shown is the result of solving for ''a'' the quadratic equation This equation can have 2, 1, or 0 positive solutions corresponding to the number of possible triangles given the data. It will have two positive solutions if only one positive solution if , and no solution if These different cases are also explained by the Side-Side-Angle congruence ambiguity. 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「law of cosines」の詳細全文を読む スポンサード リンク
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