Trigonometry/Law of Cosines

Law of Cosines Image:Law-of-cosines1.svg The Pythagorean theorem is a special case of the more general theorem relating the lengths of sides in any triangle, the law of cosines:cited work Lawrence S. Leff http://books.google.com/?id=y_7yrqrHTb4C&pg=PA326 326 0764128922 Barron's Educational Series 2005-05-01 where is the angle between sides and . Does the formula make sense? This formula had better agree with the Pythagorean Theorem when . So try it... When , The and the formula reduces to the usual Pythagorean theorem. Permutations For any triangle with angles and corresponding opposite side lengths , the Law of Cosines states that Proof Image:Law-of-cosines2.svg Dropping a perpendicular from vertex to intersect (or extended) at splits this triangle into two right-angled triangles and , with altitude from side . First we will find the lengths of the other two sides of triangle in terms of known quantities, using triangle . Side is split into two segments, with total length . has length has length Now we can use the Pythagorean Theorem to find , since . {| | | |- | | |- | | |} The corresponding expressions for and can be proved similarly. The formula can be rearranged: and similarly for and . Applications This formula can be used to find the third side of a triangle if the other two sides and the angle between them are known. The rearranged formula can be used to find the angles of a triangle if all three sides are known. See Solving Triangles Given SAS. Notes Solving Triangles Given ASA Law of Sines