13 6 The Law of Cosines Objectives Use

13 -6 The Law of Cosines Objectives: Use the Law of Cosines to solve triangles. Apply Area Formula to triangles. It doesn’t matter which side is called a, b, and c, as long as the opposite side is uppercase A, B, and C. a 2 = b 2 + c 2 - 2 b c cos A These two sides are repeated. This side is always opposite this angle. Holt Algebra 2 *Note if A = 90 , this term drops out (cos 90 = 0), and we have the normal Pythagorean theorem.

13 -6 The Law of Cosines Use these to find missing Sides. Holt Algebra 2 Use these to find missing Angles.

13 -6 The Law of Cosines Write the rule and find Angle A: (I do, you watch) Holt Algebra 2

13 -6 The Law of Cosines Write the rule and find Angle B: (Whiteboard) Holt Algebra 2

13 -6 The Law of Cosines Write the rule and find Angle C: (Whiteboard) Holt Algebra 2

13 -6 The Law of Cosines Find the measures indicated. Round to the nearest tenth. (I do, you watch) Use the Law of Cosines to set up an equation to solve for m∠C ≈ 60. 8° Instead of using the Law of Cosines, m∠B can be found using the Triangle Sum Theorem since two of the angles are now known. m∠B ≈ 39. 2° Holt Algebra 2

13 -6 The Law of Cosines Find m. Holt Algebra 2

13 -6 The Law of Cosines Find b. b ≈ 6. 8 Holt Algebra 2

13 -6 The Law of Cosines Holt Algebra 2

13 -6 The Law of Cosines Holt Algebra 2

13 -6 The Law of Cosines Example 1: 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 Holt Algebra 2 Use a calculator to simplify. Solve for the positive value of c.

13 -6 The Law of Cosines Example 1 Continued Step 2 Find the measure of the smaller angle, B. Law of Sines Substitute. Solve for m Step 3 Find the third angle measure. m A 112. 4° Holt Algebra 2 B.

13 -6 The Law of Cosines Example 2: Using the Law of Cosines 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. Holt Algebra 2 B = Cos-1 (0. 2857) ≈ 73. 4°

13 -6 The Law of Cosines Example 2 Continued Use the given measurements to solve ∆ABC (nearest tenth). Step 2 Find another angle measure 72 = 82 + 92 – 2(8)(9) cos C Substitute Law of Cosines cos C = 0. 6667 Solve for cos C. m Solve for m C. C = Cos-1 (0. 6667) ≈ 48. 2° Step 3 Find the third angle measure. m A 58. 4° Holt Algebra 2

13 -6 The Law of Cosines Remember! The largest angle of a triangle is the angle opposite the longest side. When using the LAW of COSINES, find the largest angle first. When using the LAW of SINES, find the largest angle last (using the triangle sum formula) Holt Algebra 2

13 -6 The Law of Cosines Example 3 The surface of a hotel swimming pool is shaped like a triangle with sides measuring 50 m, 28 m, and 30 m. What is the area of the pool’s surface to the nearest square meter? Find the measure of the largest angle, A. 502 = 302 + 282 – 2(30)(28) cos A m A ≈ 119. 0° Law of Cosines Solve for m Holt Algebra 2 A.

13 -6 The Law of Cosines Notes 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. x = 18; y = 10; z = 9 m Z ≈ 142. 6°; m Y ≈ 19. 7°; m Z ≈ 17. 7° Holt Algebra 2