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5.4 The Law of Cosines It is possible to solve an oblique (nonright) triangle whenever the measures of the three sides or two sides and the included angle are given by using the Law of Cosines. A C c b a b C B a B A c Law of Cosines For any οABC, where a, b, and c are the lengths of the sides opposite the angles with measures A, B, and C, respectively, a2 = b2 + c2 β 2bc cos A or cos π΄ = b2 = a2 + c2 β 2ac cos B or cos π΅ = c2 = a2 + b2 β 2ab cos C or cos πΆ = π2 +π 2 βπ2 2ππ π2 +π 2 βπ2 2ππ π2 +π2 βπ 2 2ππ Given Information Appropriate Law Three sides Law of Cosines Two sides and the included angle Law of Cosines Two sides and an angle opposite one side (ambiguous case) Law of Sines One side and two angles Law of Sines Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines Page 1 Example 1: Solve each of the following triangles. Express answers to the nearest tenth of a unit. a. οABC: mοC = 100.5ο°, a = 1.2, and b = 2.6 b. οABC: mοC = 96.4ο°, a = 5.8, and b = 8.3 c. οRST: mοS = 75.8ο°, r = 51.9, and t = 36.2 Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines Page 2 Example 2: Solve each of the following triangles. Express angle measures to the nearest degree. a. οABC: a = 14.3, b = 10.6, and c = 8.4 b. οPQR: p = 17.2, q = 15.4, and r = 13.6 c. οRST: r = 27.8, s = 31.3, and t = 23.5 Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines Page 3 Example 3: Use the law of cosines to find each solution. a. Two airplanes leave an airport at the same time. The heading of the first is 120ο° and the heading of the second is 320ο°. If the planes travel at rates of 700 and 600 mi/h, respectively, how far apart are they after 1 h? b. Two airplanes leave an airport at the same time. The heading of the first is 150ο° and the heading of the second is 260ο°. If the planes travel at rates of 680 and 560 mi/h, respectively, how far apart are they after 2 h? Homework: p. 268 => Class Exercises 1 - 8 pp. 269 β 270 => Practice Exercises 1 β 10; 14; 17; 18; 21; 22; 28; 29; 32 Advanced Mathematics/Trigonometry: Lesson 5.4 The Law of Cosines Page 4