794 CHAPTER 8 Applications of Trigonometry Recall the double-angle identity for sin 2u. sin 2u = 2 sin u cos u sin A = 2 sin a A 2b cos a A 2b Let u = A 2 . 2 bc = 2 sin a A 2b cos a A 2b Use equation (11). 2 bc = 2 B1s - b21s - c2 bc # Bs1s - a2 bc Use equations (9) and (10). 2 bc = 2 Bs1s - a21s - b21s - c2 b2c 2 Multiply. 2 bc = 22s1s - a21s - b21s - c2 bc Simplify the denominator. =!s1s −a2 1s −b2 1s −c2 Multiply by bc. Divide by 2. Heron’s formula results. 8.2 Exercises CONCEPT PREVIEW Assume a triangle ABC has standard labeling. (a) Determine whether SAA, ASA, SSA, SAS, or SSS is given. (b) Determine whether the law of sines or the law of cosines should be used to begin solving the triangle. 1. a, b, and C 2. A, C, and c 3. a, b, and A 4. a, B, and C 5. A, B, and c 6. a, c, and A 7. a, b, and c 8. b, c, and A 458 1 4√2 10. 608 8 3 Find the length of the remaining side of each triangle. Do not use a calculator. 9. Find the measure of u in each triangle. Do not use a calculator. 11. 3 7 5 u 12. 1 u 1 3 Solve each triangle. Approximate values to the nearest tenth. 13. C A B 3 5 1218 14. A 6 4 C B 618
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