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THE LAW OF SINES 2.3 I can solve triangles using the Law of Sines If none of the angles of a triangle is a right angle, the triangle is called oblique. All angles are acute Two acute angles, one obtuse angle To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles. FOUR CASES CASE 1: One side and two angles are known (SAA or ASA). CASE 2: Two sides and the angle opposite one of them are known (SSA). Ambiguous Case CASE 3: Two sides and the included angle are known (SAS). CASE 4: Three sides are known (SSS). A S A ASA CASE 1: ASA or SAA Use Law of Sines S A A SAA S A S CASE 2: SSA - Ambiguous Case Use Law of Sines S A S CASE 3: SAS Use Law of Cosine S S S CASE 4: SSS Use Law of Cosines Theorem Law of Sines b 10 sin 10 sin 70 9.40 c 10 sin 10 sin 80 9.85 12 sin 20 a 4.17 sin 100 12 sin 60 b 10.55 sin 100 The Ambiguous Case: Case 2: SSA The known information may result in One triangle Two triangles No triangles Not possible, so there is only one triangle! sin 132.5 a5 7.37 sin 30 a 7.37, b 5, c 3, 30, 17.5, 132.5 10 sin 45 sin 0.88 8 62.1 or 117.9 1 2 Two triangles!! Triangle 1: 62.1 180 45 62.1 72.9 1 1 8 sin 72.9 a1 10.81 sin 45 62.1, 72.9, 45 1 1 a1 10.81, b 8, c 10 Triangle 2: 117.9 180 45 117.9 17.1 2 1 sin 17.1 sin 45 a2 8 8 sin 17.1 a2 3.33 sin 45 117.9, 17.1, 45 2 2 a2 3.33, b 8, c 10 sin 1.28 No triangle with the given measurements! The Ambiguous Case: Case 2: SSA The known information may result in One triangle Two triangles No triangles The key to determining the possible triangles, if any, lies primarily with the height, h and the fact h = b sin α b α h a No Triangle If a < h = b sin α, then side a is not sufficiently long to form a triangle. b α a h = b sinα a < h = b sin α One Right Triangle If a = h = b sin α, then side a is just long enough to form a triangle. b α a h = b sinα a = h = b sin α Two Triangles If a < b and h = b sin α < a, then two distinct triangles can be formed b α a a h = b sinα a < b and h = b sin α < a One Triangle If a ≥ b, then only one triangle can be formed. b α a h = b sinα a≥b Fortunately we do not have to rely on the illustration to draw a correct conclusion. The Law of Sines will help us.