EQ What is the law of cosines and

EQ: What is the law of cosines, and how can we use it to solve right triangles? Warm Up Find each measure to the nearest tenth. 1. m y 104° 3. Y ≈ 18. 3 2. x ≈ 8. 8

EQ: What is the law of cosines, and how can we use it to solve right triangles? In the previous lesson, you learned to solve triangles by using the Law of Sines. However, the Law of Sines cannot be used to solve triangles for which side-angle-side (SAS) or side-side (SSS) information is given. Instead, you must use the Law of Cosines.

EQ: What is the law of cosines, and how can we use it to solve right triangles? To derive the Law of Cosines, draw ∆ABC with altitude BD. If x represents the length of AD, the length of DC is b – x.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Write an equation that relates the side lengths of ∆DBC. a 2 = (b – x)2 + h 2 Pythagorean Theorem a 2 = b 2 – 2 bx + x 2 + h 2 Expand (b – x)2. 2 = x 2 + h 2. In ∆ABD, c = – 2 bx + Substitute c 2 for x 2 + h 2. a 2 = b 2 – 2 b(c cos A) + c 2 In ∆ABD, cos A = or x = cos A. Substitute 2 2 2 a = b + c – 2 bccos A c cos A for x. a 2 b 2 c 2 The previous equation is one of the formulas for the Law of Cosines.

EQ: What is the law of cosines, and how can we use it to solve right triangles?

EQ: What is the law of cosines, and how can we use it to solve right triangles? Using the Law of Cosines Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 8, b = 5, m C = 32. 2° Step 1 Find the length of the third side. c 2 = a 2 + b 2 – 2 ab cos C Law of Cosines c 2 = 82 + 52 – 2(8)(5) cos 32. 2° Substitute. c 2 ≈ 21. 3 c ≈ 4. 6 Use a calculator to simplify. Solve for the positive value of c.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Step 2 Find the measure of the smaller angle, B. Law of Sines Substitute. Solve for sin B. Solve for m B.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Step 3 Find the third angle measure. m A + 35. 4° + 32. 2° 180° m A 112. 4° Triangle Sum Theorem Solve for m A.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 8, b = 9, c = 7 Step 1 Find the measure of the largest angle, B. b 2 = a 2 + c 2 – 2 ac cos B Law of cosines 92 = 82 + 72 – 2(8)(7) cos B = 0. 2857 Substitute. Solve for cos B. m Solve for m B. B = Cos-1 (0. 2857) ≈ 73. 4°

EQ: What is the law of cosines, and how can we use it to solve right triangles? Use the given measurements to solve ∆ABC. Round to the nearest tenth. Step 2 Find another angle measure c 2 = a 2 + b 2 – 2 ab cos C Law of cosines 72 = 82 + 92 – 2(8)(9) cos C Substitute. cos C = 0. 6667 Solve for cos C. m Solve for m C. C = Cos-1 (0. 6667) ≈ 48. 2°

EQ: What is the law of cosines, and how can we use it to solve right triangles? Use the given measurements to solve ∆ABC. Round to the nearest tenth. Step 3 Find the third angle measure. m A + 73. 4° + 48. 2° 180° m A 58. 4° Triangle Sum Theorem Solve for m A.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Use the given measurements to solve ∆ABC. Round to the nearest tenth. b = 23, c = 18, m A = 173° Step 1 Find the length of the third side. a 2 = b 2 + c 2 – 2 bc cos A Law of Cosines a 2 = 232 + 182 – 2(23)(18) cos 173° Substitute. a 2 ≈ 1672. 8 a ≈ 40. 9 Use a calculator to simplify. Solve for the positive value of c.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Step 2 Find the measure of the smaller angle, C. Law of Sines Substitute. Solve for sin C. m C = Sin-1 Solve for m C.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Step 3 Find the third angle measure. m B + 3. 1° + 173° 180° m B 3. 9° Triangle Sum Theorem Solve for m B.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 35, b = 42, c = 50. 3 Step 1 Find the measure of the largest angle, c 2 = a 2 + b 2 – 2 ab cos C C. Law of cosines 50. 32 = 352 + 422 – 2(35)(50. 3) cos C Substitute. cos C = 0. 1560 Solve for cos C. m C = Cos-1 (0. 1560) ≈ 81. 0° Solve for m C.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Use the given measurements to solve ∆ABC. Round to the nearest tenth. a = 35, b = 42, c = 50. 3 Step 2 Find another angle measure a 2 = c 2 + b 2 – 2 cb cos A 352 = 50. 32 + 422 – 2(50. 3)(42) cos A Law of cosines Substitute. cos A = 0. 7264 Solve for cos A. m Solve for m A. A = Cos-1 (0. 7264) ≈ 43. 4°

EQ: What is the law of cosines, and how can we use it to solve right triangles? Step 3 Find the third angle measure. m B + 81° + 43. 4° 180° m B 55. 6° Solve for m B.

EQ: What is the law of cosines, and how can we use it to solve right triangles? Lesson Quiz Use the given measurements to solve ∆ABC. Round to the nearest tenth. 1. a = 18, b = 40, m C = 82. 5° c ≈ 41. 7; m A ≈ 25. 4°; m B ≈ 72. 1° 2. a = 18. 0; b = 10; c = 9 m A ≈ 142. 6°; m B ≈ 19. 7°; m C ≈ 17. 7°