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Section 9.4: The Law of Cosines The Law of Cosines We use the Law of cosines to solve triangles that are not right-angled. In particular, when we know two sides of a triangle and their included angle, then the Law of Cosines enables us to find the third side. Thus if we know sides a and b and their included angle θ, then the Law of Cosines states: c² = a² + b² − 2ab cos θ The Law of Cosines is a extension of the Pythagorean theorem; because if θ were a right angle, we would have c² = a² + b². Example 1. In triangle DEF, side e = 8 cm, f = 10 cm, and the angle at D is 60°. Find side d. d² = e² + f² − 2ef cos 60° d² = 8² + 10² − 2· 8· 10· ½, since cos 60° = ½, d² = 164 − 80 d² = 84 d = Example 2: In triangle PQR, find side r if side p = 5 in, q = 10 in, and the included angle of 14°. r² = 5² + 10² − 2· 5· 10 cos 14° r² = 25 + 100 – 100 (cos 14°) r² ≈ 27.97 (store decimal in calc.) r ≈ 5.3 in. If we solve the law of cosines for cos C, we obtain: cos C = a2 + b2 – c2 2ab In this form, the law of cosines can be used to find the measures of the angles of a triangle when the lengths of three sides are known. Example 3: A triangle has sides of lengths 6, 12, and 15. a. Find the measure of the smallest angle. b. Find the length of the median to the longest side. B 6 A 12 7.5 D 7.5 C a. The smallest angle of triangle ABC is opposite the shortest side, AB. cos C = 122 + 152 – 62 2 · 12 · 15 = 0.925 C = Cos-1 0.925 ≈ 22.3° b. In triangle BCD, (use the law of cosines) (BD)2 = 122 + (7.5)2 – 2(12)(7.5)(0.925) = 33.75 ≈5.8 When do you use the Law of Sines and the Law of Cosines? Given: Use: To find: SAS Law of Cosines The third side and then one of the remaining angles. SSS Law of Cosines Any two angles. ASA or AAS Law of Sines The remaining sides. SSA Law of Sines An angle opposite a given side and then the third side. (Note that 0, 1, or 2 triangles are possible.) HOMEWORK pg. 352 – 353; 1, 2, 5, 6, 9 ASSIGNMENT pg. 353; 15, 16