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The Cosine Law

6.6 Adjusting the Pythagorean Theorem. C. c 2 = a 2 + b 2. b. a. The Cosine Law. c. B. A. 5. 3. 4. We start with a rope 12 units long. Units are marked. We form the following figure. A right angle is formed. 3, 4, 5 is a Pythagorean Triple.

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The Cosine Law

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  1. 6.6 Adjusting the Pythagorean Theorem C c2 = a2 + b2 b a The Cosine Law c B A

  2. 5 3 4 We start with a rope 12 units long. Units are marked. We form the following figure. A right angle is formed. 3, 4, 5 is a Pythagorean Triple

  3. Which numbers are Pythagorean Triples? no yes

  4. A Try the Pythagorean TheoremÐC < 90º 51º 6.9 cm 4.7 cm Compare c2 with a2 + b2 43º c2a2 + b2 86º B C 5.3 cm 6.92 5.32 + 4.72 c2 < a2 + b2

  5. Try the Pythagorean Theorem ÐC > 90º A Compare c2 with a2 + b2 46º c2a2 + b2 7.4 cm 7.42 5.32 + 4.72 4.7 cm c2 > a2 + b2 39º 95º B C 5.3 cm

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

  7. Example (1) Find a. a2 = b2 + c2 – 2bc cos A a2 = 4.62 + 6.22 – 2(4.6)(6.2) cos 52º A a2 = 21.16 + 38.44 – 35.12 52º a2 = 24.48 6.2 cm 4.6 cm a= 4.9 cm a B C

  8. b = Example (2) Find b. b2 = a2 + c2 – 2ac cos B b2 = 5.02 + 5.72 – 2(5.0)(5.7) cos 78º b2 = 25 + 32.49 – 11.85 A b2 = 45.64 b 5.7 cm 78º b = 6.8 B C 5.0 cm

  9. – 96 = cos P – 144 R Example (3) Find ÐP. p2 = q2 + r2 – 2qrcos P 8 7 72 = 82 + 92 – 2(8)(9) cos P Q 49 = 64 + 81– 144 cos P P 9 49 – 64 – 81 = – 144 cos P ÐP = 48.2º – 96 = – 144 cos P 0.6666666 = cos P ÐP = cos-1(0.66666)

  10. 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

  11. A b c In DADC h a – x x B D C a In DABD Deriving the Cosine Law: Given DABC and all angles are acute. Draw altitude AD.

  12. A ÐA = cos-1(0.0370) Given: SSS (finding the angle) Ex:Find the largest angle 15 18 Find Ð A. a2 = b2 + c2 – 2bccos A B C 23 232 = 182 + 152 – 2(18)(15) cos A 529 = 324 + 225– 540 cos A 540 cos A = 324 + 225– 529 cos A = 0.0370 ÐA = 87.9°

  13. b= B Ex: If a = 39 cm, Ð B = 48° and c = 57 cm, 48º 57 cm Find b. 39 cm b2 = a2 + c2 – 2ac cos B A C b b2 = 392 + 572 – 2(39)(57) cos 48º b2 = 1521 + 3249– 2975 b2 = 1794 b = 42.4 cm

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