SECTION 8.3 The Law of Cosines 571 Figure 36 608 2 3 B A c 1 Solve SAS Triangles The Law of Cosines is used to solve Case 3 (SAS), which applies to triangles for which two sides and the included angle are known. Using the Law of Cosines to Solve an SAS Triangle Solve the triangle: a b C 2, 3, 60 = = = ° EXAMPLE 1 Solution See Figure 36. Because two sides, a and b, and the included angle, C 60 , = ° are known, the Law of Cosines makes it easy to find the third side, c. c a b ab C c 2 cos 2 3 223cos60 13 12 1 2 7 7 2 2 2 2 2 = + − = + − ⋅ ⋅ ⋅ ° = − ⋅ = = a b C 2, 3, 60 = = = ° Side c is of length 7. To find the angles A and B, either the Law of Sines or the Law of Cosines may be used. It is preferable to use the Law of Cosines because it will lead to an equation with one solution whether solving for A or B. Using the Law of Sines would lead to an equation with two solutions that would need to be checked to determine which solution fits the given data.* We choose to use formulas (2) and (3) of the Law of Cosines to find A and B. For A: a b c bc A bc A b c a A b c a bc A 2 cos 2 cos cos 2 9 7 4 2 3 7 12 6 7 2 7 7 cos 2 7 7 40.9 2 2 2 2 2 2 2 2 2 1 = + − = + − = + − = + − ⋅ = = = ≈ ° − For B: b a c ac B B a c b ac B 2 cos cos 2 4 7 9 4 7 2 4 7 7 14 cos 7 14 79.1 2 2 2 2 2 2 1 = + − = + − = + − = = = ≈ ° − Notice that A B C 40.9 79.1 60 180 , ++= °+ °+°= ° as required. Now Work PROBLEMS 9 AND 17 *The Law of Sines can be used when seeking the angle opposite the smaller side, since it is acute. (In Figure 36, use the Law of Sines to find A, the angle opposite the smaller side.) 2 Solve SSS Triangles The next example uses the Law of Cosines to solve a triangle when three sides are known, Case 4 (SSS). TIP The angle B can also be found using A B C 180 , + + = ° so B 180 40.9 60 79.1 . = °− °− ° = ° j
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