SECTION 8.2 The Law of Sines 561 Now use the Law of Sines (twice) to find the unknown sides b and c. = = A a B b A a C c sin sin sin sin Because = = ° = ° a A B 4, 40 , 60 , and = ° C 80 , we have ° = ° ° = ° b c sin40 4 sin60 sin40 4 sin80 Solving for b and c yields = ° ° ≈ = ° ° ≈ b c 4sin60 sin40 5.39 4sin80 sin40 6.13 Notice in Example 1 that b and c are found by working with the given side a. This is better than finding b first and working with a rounded value of b to find c. Now Work PROBLEM 11 Figure 21 c C b 408 608 4 TIP Although it is not a complete check, the reasonableness of answers can be verified by determining whether the longest side is opposite the largest angle and the shortest side is opposite the smallest angle. j Using the Law of Sines to Solve an ASA Triangle Solve the triangle: = ° = ° = A B c 35 , 15 , 5 Solution EXAMPLE 2 Figure 22 illustrates the triangle to be solved. Two angles are known ( = ° A 35 and = ° B 15 ). Find the third angle using formula (2): + + = ° ° + ° + = ° = ° A B C C C 180 35 15 180 130 Now the three angles and one side ( ) = c 5 of the triangle are known. To find the remaining two sides a and b, use the Law of Sines (twice). = = ° = ° ° = ° = ° ° ≈ = ° ° ≈ A a C c B b C c a b a b sin sin sin sin sin 35 sin130 5 sin15 sin130 5 5 sin35 sin130 3.74 5 sin15 sin130 1.69 Figure 22 358 158 5 a b C Figure 23 = h b A sin a h A b Now Work PROBLEM 25 2 Solve SSA Triangles Case 2 (SSA), which describes triangles for which two sides and the angle opposite one of them are known, is referred to as the ambiguous case, because the known information may result in one triangle, two triangles, or no triangle at all. Suppose that sides a and b and angle A are given, as illustrated in Figure 23.The key to determining how many triangles, if any, can be formed from the given information lies primarily with the relative size of side a, the height h, and the fact that = h b A sin . Figure 24 a h b A sin < = a A b h 5 b sin A Figure 25 a h b A sin = = a A b h 5 b sin A No Triangle If < = a h b A sin , then side a is not long enough to form a triangle. See Figure 24. One Right Triangle If = = a h b A sin , then side a is just long enough to form one right triangle. See Figure 25.
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