560 CHAPTER 8 Applications of Trigonometric Functions CAUTION Oblique triangles cannot be solved using the methods of Section 8.1. Do you know why? j Using the Law of Sines to Solve an SAA Triangle Solve the triangle: = ° = ° = A B a 40 , 60 , 4 EXAMPLE 1 To solve an oblique triangle means to find the lengths of its sides and the measures of its angles. To do this, we need to know the length of one side,* along with (i) two angles, (ii) one angle and one other side, or (iii) the other two sides. There are four possibilities to consider. Case 1 One side and two angles are known (ASA or SAA). Case 2 Two sides and the angle opposite one of them are known (SSA). Case 3 Two sides and the included angle are known (SAS). Case 4 Three sides are known (SSS). Figure 20 illustrates the four cases, where the known measurements are shown in blue. The Law of Sines is used to solve triangles for which Case 1 or 2 holds. Cases 3 and 4 are considered when we study the Law of Cosines in the next section. THEOREM Law of Sines For a triangle with sides a, b, c and opposite angles A, B, C, respectively, = = A a B b C c sin sin sin (1) = = = A a B b A a C c B b C c • sin sin • sin sin • sin sin + + = ° A B C 180 (2) A proof of the Law of Sines is given at the end of this section. The Law of Sines consists of three equalities: Formula (1) is a compact way to write these three equations. To solve triangles with the Law of Sines, we use the fact that the sum of the angles of any triangle equals ° 180 ; that is, 1 Solve SAA or ASA Triangles The first two examples show how to solve a triangle when one side and two angles are known (Case 1: SAA or ASA). Figure 20 S A Case 1: ASA A S S A Case 2: SSA S A Case 1: SAA S S Case 3: SAS A A S S S Case 4: SSS Solution Figure 21 on the next page shows the triangle to be solved.The third angle C is found using formula (2). + + = ° ° + ° + = ° = ° A B C C C 180 40 60 180 80 *The length of one side must be known because knowing only the angles results in a family of similar triangles .
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