580 CHAPTER 8 Applications of Trigonometric Functions ‘Are You Prepared?’ The answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 8.4 Assess Your Understanding 1. The area K of a triangle whose base is b and whose altitude is h is . (p. A15) 2. True or False θ θ = + cos 2 1 sin 2 2 (pp. 528–530) Skill Building In Problems 9–16, find the area of each triangle. Round answers to two decimal places. 9. 458 2 4 b A C 10. 308 4 3 a B C 11. 948 5 7 c B A 12. 208 2 5 b C A 13. 7 4 9 C A B 14. 8 5 4 C B A 15. 9 6 4 C B A 16. 3 4 4 C A B Concepts and Vocabulary 3. If two sides a and b and the included angle C are known in a triangle, then the area K is found using the formula = K . 4. The area K of a triangle with sides a, b, and c is = K , where = s . 5. Find the area of the right triangle whose legs are of length 3 and 4. 6. True or False The area of a triangle equals one-half the product of the lengths of two of its sides times the sine of their included angle. 7. Multiple Choice Given two sides of a triangle, b and c, and the included angle A, the altitude h from angle B to side b is given by . (a) ab A 1 2 sin (b) b A sin (c) c A sin (d) bc A 1 2 sin 8. Multiple Choice Heron’s Formula is used to find the area of triangles. (a) ASA (b) SAS (c) SSS (d) AAS In Problems 17–28, find the area of each triangle. Round answers to two decimal places. 17. = = = ° a b C 3, 4, 50 18. = = = ° a c B 2, 1, 10 19. = = = ° b c A 1, 8, 75 20. = = = ° a b C 6, 4, 60 21. = = = ° a c B 3, 2, 115 22. = = = ° b c A 4, 1, 120 23. = = = a b c 12, 35, 37 24. = = = a b c 4, 5, 3 25. = = = a b c 4, 4, 4 26. = = = a b c 3, 3, 2 27. = = = a b c 11, 14, 20 28. = = = a b c 4, 3, 6 Applications and Extensions 29. Area of an ASA Triangle If two angles and the included side are given, the third angle is easy to find. Use the Law of Sines to show that the area K of a triangle with side a and angles A, B, and C is = K a B C A sin sin 2 sin 2 30. Area of a Triangle Prove the two other forms of the formula for the area K of a triangle given in Problem 29. = = K b A C B K c A B C sin sin 2sin and sin sin 2sin 2 2 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure
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