781 8.1 The Law of Sines Next use the law of sines to find side a. a sin A = b sin B Law of sines a sin 24° 40′ = 27.3 sin 102° 40′ Substitute known values. a = 27.3 sin 24° 40′ sin 102° 40′ Multiply by sin 24° 40′. a ≈11.7 cm Use a calculator. Now that we know two sides, a and b, and their included angle C, we find the area. = 1 2 ab sin C≈ 1 2 111.72127.32 sin 52° 40′ ≈127 cm2 S Now Try Exercise 81. Solve for a. 11.7 is an approximation for a. In practice, use the calculator value. 8.1 Exercises CONCEPT PREVIEW Consider each case and determine whether there is sufficient information to solve the triangle using the law of sines. 1. Two angles and the side included between them are known. 2. Two angles and a side opposite one of them are known. 3. Two sides and the angle included between them are known. 4. Three sides are known. 5. CONCEPT PREVIEW Which one of the following sets of data does not determine a unique triangle? A. A = 50°, b = 21, a = 19 B. A = 45°, b = 10, a = 12 C. A = 130°, b = 4, a = 7 D. A = 30°, b = 8, a = 4 6. CONCEPT PREVIEW Which one of the following sets of data determines a unique triangle? A. A = 50°, B = 50°, C = 80° B. a = 3, b = 5, c = 20 C. A = 40°, B = 20°, C = 30° D. a = 7, b = 24, c = 25 CONCEPT PREVIEW In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following? (a) two triangles (b) exactly one triangle (c) no triangle CONCEPT PREVIEW Determine the number of triangles ABC possible with the given parts. 9. a = 50, b = 26, A = 95° 10. a = 31, b = 26, B = 48° 11. c = 50, b = 61, C = 58° 12. a = 35, b = 30, A = 40° 5 (–3, 4) x 0 y 7. 5 (3, 4) x 0 y 8.
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