790 CHAPTER 8 Applications of Trigonometry CAUTION Had we used the law of sines to find C rather than B in Example 2, we would not have known whether C was equal to 81.7° or to its supplement, 98.3°. EXAMPLE 3 Applying the Law of Cosines (SSS) Solve triangle ABC if a = 9.47 ft, b = 15.9 ft, and c = 21.1 ft. SOLUTION We can use the law of cosines to solve for any angle of the triangle. We solve for C, the largest angle. We will know that C is obtuse if cos C60. c2 = a2 + b2 - 2ab cos C Law of cosines cos C = a2 + b2 - c2 2ab Solve for cos C. cos C = 9.472 + 15.92 - 21.12 219.472115.92 Substitute. cos C≈ -0.34109402 Use a calculator. C≈109.9° Use the inverse cosine function. Now use the law of sines to find angle B. sin B b = sin C c Law of sines (alternative form) sin B 15.9 = sin 109.9° 21.1 Substitute. sin B = 15.9 sin 109.9° 21.1 Multiply by 15.9. B ≈45.1° Use the inverse sine function. Since A = 180° - B - C, we have A ≈180° - 45.1° - 109.9° ≈25.0°. S Now Try Exercise 23. Trusses are frequently used to support roofs on buildings, as illustrated in Figure 16. The simplest type of roof truss is a triangle, as shown in Figure 17. (Data from Riley, W., L. Sturges, and D. Morris, Statics and Mechanics of Materials, John Wiley and Sons.) Figure 16 EXAMPLE 4 Designing a RoofTruss (SSS) Find angle B to the nearest degree for the truss shown in Figure 17. SOLUTION b2 = a2 + c2 - 2ac cos B Law of cosines cos B = a2 + c2 - b2 2ac Solve for cos B. cos B = 112 + 92 - 62 21112192 Let a = 11, b = 6, and c = 9. cos B ≈0.83838384 Use a calculator. B ≈33° Use the inverse cosine function. S Now Try Exercise 49. 9 ft 6 ft 11 ft A B C Figure 17
RkJQdWJsaXNoZXIy NjM5ODQ=