The Cosine Law The Cosine Law is used for situations involving SAS as well as SSS. You are given 2 sides and the contained angle and you wish to find the.

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The Law of Cosines The law of sines cannot be used to solve a triangle where two sides and the included angle (SAS) are given or three sides (SSS) are.

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The Cosine Law The Cosine Law is used for situations involving SAS as well as SSS. You are given 2 sides and the contained angle and you wish to find the third side or three side and you need to find one of the angles. a 2 = b 2 + c 2 – 2bc cos A c 2 = a 2 + b 2 – 2ab cos C b 2 = a 2 + c 2 – 2ac cos B A B C c a b

Using the Cosine Law to Find an Angle c 2 = a 2 + b 2 – 2ab cos C cos C = a 2 + b 2 – c 2 2ab 2ab cos C = a 2 + b 2 – c 2

Example (1) Find a. a 2 = a = 4.9 cm a 2 = – 2(4.6)(6.2) cos 52º 6.2 cm 4.6 cm A BC 52º a a 2 = b 2 + c 2 – 2bc cos A

Example (2) Find b. b 2 = – 2(5.0)(5.7) cos 78º b 2 = a 2 + c 2 – 2ac cos B b 2 = b = b = cm 5.0 cm A B C 78º b

Example (3) Find  P. 7 P Q R 8 9 cos P =  P = cos -1 ( )  P = 48.2º

Example 4 41º 16 km/h 14 km/h Two girls begin cycling from the same location. The angle of the roads is 41º. One girl is cycling at 14 km/h and the second girl is cycling at 16 km/h. How far apart are the girls after 3 hours? 41º 48 km 42 km x x 2 = – 2(42)(48)cos 41º 3 hours x 2 = 1025

Given: SSS (finding the angle) Find  A. B C A cos A =  A = 87.9°  A = cos -1 (0.0370) Ex: Find the largest angle cos A = (15)(18)

Find b. b = 42.4 cm b 2 = 1794 b 2 = – 2(39)(57) cos 48º b =b = C B A 48º 57 cm 39 cm b Ex: If a = 39 cm,  B = 48° and c = 57 cm,