774 CHAPTER 8 Applications of Trigonometry Using the Law of Sines EXAMPLE 1 Applying the Law of Sines (SAA) Solve triangle ABC if A = 32.0°, B = 81.8°, and a = 42.9 cm. SOLUTION Start by drawing a triangle, roughly to scale, and labeling the given parts as in Figure 2. The values of A, B, and a are known, so use the form of the law of sines that involves these variables, and then solve for b. a sin A = b sin B 42.9 sin 32.0° = b sin 81.8° Substitute the given values. b = 42.9 sin 81.8° sin 32.0° Multiply by sin 81.8° and rewrite. b ≈80.1 cm Approximate with a calculator. To find C, use the fact that the sum of the angles of any triangle is 180°. A + B + C = 180° Angle sum formula C = 180° - A - B Solve for C. C = 180° - 32.0° - 81.8° Substitute. C = 66.2° Subtract. Now use the law of sines to find c. (The Pythagorean theorem does not apply because this is not a right triangle.) a sin A = c sin C Law of sines 42.9 sin 32.0° = c sin 66.2° Substitute known values. c = 42.9 sin 66.2° sin 32.0° Multiply by sin 66.2° and rewrite. c ≈74.1 cm Approximate with a calculator. S Now Try Exercise 19. Choose a form of the law of sines that has the unknown variable in the numerator. CAUTION Whenever possible, use given values in solving triangles, rather than values obtained in intermediate steps, to avoid rounding errors. EXAMPLE 2 Applying the Law of Sines (ASA) An engineer wishes to measure the distance across a river. See Figure 3. He determines that C = 112.90°, A = 31.10°, and b = 347.6 ft. Find the distance a. SOLUTION To use the law of sines, one side and the angle opposite it must be known. Here b is the only side whose length is given, so angle B must be found before the law of sines can be used. B = 180° - A - C Angle sum formula, solved for B B = 180° - 31.10° - 112.90° Substitute the given values. B = 36.00° Subtract. b = 347.6 ft 31.108 A a C B 112.908 Figure 3 A b c a 32.08 81.88 42.9 cm C B Figure 2 Be sure to label a sketch carefully to help set up the correct equation.
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