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Geometry Section 9.7 The Law of Sines What you will learn: 1. Find areas of triangles 2. Use the Law of Sines to solve triangles. 3. Use the Law of Cosines to solve triangles Example: Find the area of each triangle. The 1 bh formula for the area of a triangle is ________ 2 h h sin 56 8 8 sin 56 h h 6.632 A 1 12 6.632 39.792 2 h 70 sin 70 h 12 12 sin 70 h h 11.276 A 1 15 11.276 84.57 2 In sections 9.4 - 9.6, we learned how to use trigonometry to solve right triangles. The Law of Sines allows us to find sides and/or angles in any triangle. Theorem 9.9: Law of Sines sin A a sin B b sin C c Examples: Solve each triangle. Round to the nearest 1000th as necessary. BC sin 24 2 sin 56 100 2 sin 56 BC sin 24 4.077 AB sin 24 2 sin 100 mC 180 56 24 100 sin 24 sin 56 sin 100 2 BC AB 2 sin 100 AB sin 24 4.842 Examples: Solve each triangle. Round to the nearest 1000th as necessary. sin 29 sin A sin C 2 2.34 AB 34.557 2 sin A 2.34 sin 29 mC 180 29 34.557 116.443 2.34 sin 29 sin A 2 1 2.34 sin 29 A sin 2 AB sin 29 2 sin 116.443 34.557 2 sin 116.443 AB sin 29 3.694 Note that you cannot, at least initially, apply the Law of Sines to solve the triangle at the right. Why not. sin 40 sin A sin B c 20 25 To use the Law of Sines you must know an angle AND the opposite side. Theorem 9.10: Law of Cosines For any triangle ABC with sides a, b and c opposite angles A, B and C respectively, a b c 2bc cos A 2 2 2 b a c 2ac cos B 2 2 2 c a b 2ab cos C 2 2 2 Example: Solve the triangle at the top right. c a b 2ab cos C 2 2 2 c 20 25 2(20)(25) cos 40 2 2 2 c 258.9555569 c 16.092 2 sin 40 sin A sin B 16.092 20 25 sin 40 sin A 16.092 20 16.092 sin A 20 sin 40 20 sin 40 sin A 16.092 20 sin 40 A sin 53.024 16.092 1 B 180 40 53.024 86.976 Example: Solve the triangle at the right. NOTE: When given the 3 sides of a triangle, you should always solve for the largest angle first. b a c 2ac cos B 2 2 2 27 12 20 2(12)(20) cos B 729 144 400 480 cos B 144 400 1 185 B cos 480 185 480 cos B 2 2 2 185 cos B 480 B 112.670 Example: Solve the triangle at the right. NOTE: When given the 3 sides of a triangle, you should always solve for the largest angle first. sin C sin A sin 122.670 20 12 27 sin A sin 122.670 12 27 27 sin A 12 sin 122.670 12 sin 122.670 sin A 27 12 sin 122.670 A sin 27 1 A 21.971 C 180 122.670 21.971 35.359 HW: pp513 – 514 / 7-10, 13, 14, 20, 21, 28, 30, 32