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The Cosine Law

C. b. a. c. B. A. The Cosine Law. a 2 = b 2 + c 2 – 2 bc cos A. b 2 = a 2 + c 2 – 2 ac cos B. c 2 = a 2 + b 2 – 2 ab cos C. The Cosine Law is used for situations involving SAS as well as SSS.

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The Cosine Law

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  1. C b a c B A The Cosine Law a2 = b2 + c2 – 2bc cos A b2 = a2 + c2 – 2ac cos B c2 = a2 + b2 – 2ab cos C 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.

  2. Using the Cosine Law to Find an Angle c2 = a2 + b2 – 2ab cos C 2abcos C = a2 + b2 – c2 cos C = a2 + b2 – c2 2ab

  3. Example (1) Find a. a2 = b2 + c2 – 2bc cos A a2 = 4.62 + 6.22 – 2(4.6)(6.2) cos 52º A a2 = 24.48 52º 6.2 cm 4.6 cm a= 4.9 cm a B C

  4. b = Example (2) Find b. b2 = a2 + c2 – 2ac cos B b2 = 5.02 + 5.72 – 2(5.0)(5.7) cos 78º b2 = 45.64 A b 5.7 cm b = 6.8 78º B C 5.0 cm

  5. R Example (3) Find ÐP. 8 7 Q P 9 ÐP = 48.2º cos P = 0.6666666 ÐP = cos-1(0.66666)

  6. Example 4 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? 42 km 14 km/h x 3 hours 41º 41º 16 km/h 48 km x2 = 422 + 482 – 2(42)(48)cos 41º x2 = 1025

  7. A ÐA = cos-1(0.0370) Given: SSS (finding the angle) Ex:Find the largest angle 15 18 Find Ð A. B C 23 cos A = 152 + 182 - 232 2(15)(18) cos A = 0.0370 ÐA = 87.9°

  8. b= B Ex: If a = 39 cm, Ð B = 48° and c = 57 cm, 48º 57 cm Find b. 39 cm A C b b2 = 392 + 572 – 2(39)(57) cos 48º b2 = 1794 b = 42.4 cm

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