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Chapter 7 Applications of Trigonometric Functions © 2011 Pearson Education, Inc. All rights reserved © 2010 2011 Pearson Education, Inc. All rights reserved 1 SECTION 7.3 The Law of Cosines OBJECTIVES 1 2 3 Derive the Law of Cosines. Use the Law of Cosines to solve SAS triangles. Use the Law of Cosines to solve SSS triangles. THE LAW OF COSINES In triangle ABC, with sides of lengths a, b, and c, a 2 b 2 c 2 2bc cos A, b c a 2ca cos B, and 2 2 2 c a b 2ab cos C. 2 2 2 In words, the square of any side of a triangle is equal to the sum of the squares of the length of the other two sides less twice the product of the lengths of the other sides and the cosine of their included angle. © 2011 Pearson Education, Inc. All rights reserved 3 DERIVATION OF THE LAW OF COSINES a d C , B a 2 d C , B 2 a 2 b cos A c b sin A 0 2 2 a 2 b 2 cos 2 A 2bc cos A c 2 b 2 sin 2 A a 2 b 2 c 2 2bc cos A © 2011 Pearson Education, Inc. All rights reserved 4 SOLVING SAS TRIANGLES Step 1 Use the appropriate form of the Law of Cosines to find the side opposite the given angle. Step 2 Use the Law of Sines to find the angle opposite the shorter of the two given sides. Note that this angle is always an acute angle. Step 3 Use the angle sum formula to find the third angle. Step 4 Write the solution. © 2011 Pearson Education, Inc. All rights reserved 5 EXAMPLE 1 Solving SAS Triangles Solve triangle ABC with a = 15 inches, b = 10 inches, and C = 60º. Round each answer to the nearest tenth. Solution Step 1 Find side c opposite the given angle C. c 2 a 2 b 2 2ab cos C c 15 10 2 15 10 cos 60° 2 2 2 1 c 225 100 2 15 10 175 2 2 c 175 13.2 © 2011 Pearson Education, Inc. All rights reserved 6 EXAMPLE 1 Solving the SAS Triangles Solution continued Step 2 Find the angle B opposite side b. sin B sin C b c b sin C sin B c 10sin 60° sin B 175 1 10sin 60° B sin 40.9° 175 © 2011 Pearson Education, Inc. All rights reserved 7 EXAMPLE 1 Solving the SAS Triangles Solution continued Step 3 Use the angle sum formula to find the third angle. A 180° 60° 40.9° 79.1° Step 4 The solution of triangle ABC is A ≈ 79.1° a = 15 inches B ≈ 40.9° b = 10 inches C = 60° c ≈ 13.2 inches © 2011 Pearson Education, Inc. All rights reserved 8 EXAMPLE 2 Using the Law of Cosines Suppose that a Boeing 747 is flying over Disney World headed due south at 552 miles per hour. Twenty minutes later, an F-16 passes over Disney World with a bearing of N 37º E at a speed of 1250 mi/hr. Find the distance between the two planes three hours after the F-16 passes over Disney World. Round the answer to the nearest tenth. © 2011 Pearson Education, Inc. All rights reserved 9 EXAMPLE 2 Using the Law of Cosines Solution Suppose the F-16 has been traveling for t hours after passing over Disney World. Then, because the Boeing 747 had a head start of 1 20 minutes = hour, the 3 Boeing 747 has been 1 traveling t hours 3 due south. The distance between the two planes is d. © 2011 Pearson Education, Inc. All rights reserved 10 EXAMPLE 2 Using the Law of Cosines Solution continued Using the Law of Cosines in triangle FDB, we have d 1250t 2 2 2 1 1 552 t 2 1250t 552 t cos143º. 3 3 Substitute t = 3. d 28, 469, 270.04 2 d 5335.7 miles © 2011 Pearson Education, Inc. All rights reserved 11 SOLVING SSS TRIANGLES Step 1 Use the Law of Cosines to find the angle opposite the given side. Step 2 Use the Law of Sines to find either of the two remaining acute angles. Step 3 Use the angle sum formula to find the third angle. Step 4 Write the solution. © 2011 Pearson Education, Inc. All rights reserved 12 EXAMPLE 3 Solving the SSS Triangles Solve triangle ABC with a = 3.1 feet, b = 5.4 feet, and c = 7.2 feet. Round answers to the nearest tenth. Solution Step 1 Because c is the longest side, find C. c 2 a 2 b 2 2ab cos C 2ab cos C a 2 b 2 c 2 3.1 5.4 7.2 a b c cos C 0.39 2ab 2 3.1 5.4 2 2 2 2 2 2 C cos 1 0.39 113° © 2011 Pearson Education, Inc. All rights reserved 13 EXAMPLE 3 Solving the SSS Triangles Solution continued Step 2 Find B. sin B sin C b c b sin C sin B c 1 b sin C B sin c 1 5.4sin113° sin 43.7° 7.2 © 2011 Pearson Education, Inc. All rights reserved 14 EXAMPLE 3 Solving the SSS Triangles Solution continued Step 3 A ≈ 180º − 43.7º − 113º ≈ 23.3º Step 4 Write the solution. A ≈ 23.3° a = 3.1 feet B ≈ 43.7° b = 5.4 feet C ≈ 113° c = 7.2 feet © 2011 Pearson Education, Inc. All rights reserved 15 EXAMPLE 4 Solving an SSS Triangle Solve triangle ABC with a = 2 meters, b = 9 meters, and c = 5 meters. Round each answer to the nearest tenth. Solution Find B, the angle opposite the longest side. b2 c 2 a 2 2ca cos B c2 a 2 b2 52 2 2 9 2 cos B 2ca 25 2 cos B 2.6 The range of the cosine function is [–1, 1]; there is no angle B with cos B = −2.6; the triangle cannot exist. © 2011 Pearson Education, Inc. All rights reserved 16