776 CHAPTER 8 Applications of Trigonometry Description of the Ambiguous Case We have used the law of sines to solve triangles involving Case 1, given SAA or ASA. If we are given the lengths of two sides and the angle opposite one of them (Case 2, SSA), then zero, one, or two such triangles may exist. (There is no SSA congruence axiom.) Suppose we know the measure of acute angle A of triangle ABC, the length of side a, and the length of side b. We must draw the side of length a opposite angle A. The table shows possible outcomes. This situation (SSA) is called the ambiguous case of the law of sines. As shown in the table, if angle A is acute, there are four possible outcomes. If A is obtuse, there are two possible outcomes. Possible Outcomes for Applying the Law of Sines Angle A is Possible Number of Triangles Sketch Applying Law of Sines Leads to Acute 0 A C b a h sin B71, a 6h 6b Acute 1 A C b B a = h sin B = 1, a = h and h 6b Acute 1 A C b a B h 0 6sin B61, a Ú b Acute 2 A C b a B1 B2 a h 0 6sin B1 61, h 6a 6b, A + B2 6180° Obtuse 0 a C b A sin B Ú 1, a … b Obtuse 1 a C b A B 0 6sin B61, a 7b The following basic facts help determine which situation applies. Applying the Law of Sines 1. For any angle u of a triangle, 0 6sin u … 1. If sin u = 1, then u = 90° and the triangle is a right triangle. 2. sin u = sin1180° - u2 (Supplementary angles have the same sine value.) 3. The smallest angle is opposite the shortest side, the largest angle is opposite the longest side, and the middle-valued angle is opposite the intermediate side (assuming the triangle has sides that are all of different lengths).
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