Exercise Set 7.1: Solving Right Triangles For each of the following examples, (a) Write a trigonometric equation that can be used to find x, and write a trigonometric equation that can be used to find y. Each equation should involve the given acute angle and the given side length from the diagram. (Do not solve the equations in this step.) (b) Solve the equations from part (a) to find exact values for x and y. 6. y x 30o 15 7. x 7 2 45 (c) As an alternative solution to the method above, use properties of special right triangles (from Section 4.1) to find exact values of x and y. 1. 10 y y 8. 45o 11 x 30o x y 2. 45o y x Use a calculator to evaluate the following. Round answers to the nearest ten-thousandth. ( ) (b) cos 73 10. (a) cos 16 ( ) (b) tan 67 11. (a) tan 9 2 (b) sin 7 12. (a) sin 10 3 (b) cos 11 9. 7 3. y 7 3 60o x 4. ( ) ( ) In the equations below, x represents an acute angle of a triangle. Use a calculator to solve for x. Round answers to the nearest hundredth of a degree. x y (a) sin 48 13. sin ( x ) = 0.7481 60o 9 14. cos ( x ) = 0.0256 5. 15. tan ( x ) = 4 y 5 45o 16. tan ( x ) = 8 15 x Math 1330, Precalculus The University of Houston Chapter 7: Trigonometric Applications to Triangles Exercise Set 7.1: Solving Right Triangles 17. cos ( x ) = 16 17 18. sin ( x ) = 4 7 25. In GEO with right angle E, GE = 9 cm and OE = 7 cm . Find G . 26. In CAR with right angle R, CA = 11 in and AR = 8 in . Find A . Find the indicated side or angle measures (where angles are measured in degrees). Round answers to the nearest hundredth. 19. Find x. Solve each of the following triangles, i.e. find all missing side and angle measures (where angles are measured in degrees). Round all answers to the nearest hundredth. J 27. 23 38o 3 x 61 M 20. Find x. x D 54o 28. C 14 7 21. In ABC with right angle C, A = 74 and BC = 32 in . Find AC . U B 12 22. In QRS with right angle R, QR = 7 in and Q = 12 . Find QS . 29. P 8 3 23. Find H . H H L 17 13 I J T 30. 8 24. Find . 47 4 I R 3 Math 1330, Precalculus The University of Houston Chapter 7: Trigonometric Applications to Triangles Exercise Set 7.1: Solving Right Triangles For each of the following examples, (a) Draw a diagram to represent the given situation. (b) Find the indicated measure, to the nearest tenth. 31. An isosceles triangle has sides measuring 9 inches, 14 inches, and 14 inches. What are the measures of its angles? 32. An isosceles triangle has sides measuring 10 inches, 7 inches, and 7 inches. What are the measures of its angles? 33. An 8-foot ladder is leaned against the side of the house. If the ladder makes a 72 angle with the ground, how high up does the ladder reach? 34. A 14-foot ladder is leaned against the side of the house. If the ladder makes a 63 angle with the ground, how high up does the ladder reach? 35. The angle of elevation to the top of a flagpole from a point on the ground 40 ft from the base of the flagpole is 51 . Find the height of the flagpole. 36. The angle of elevation to the top of a flagpole from a point on the ground 18 meters from the base of the flagpole is 43 . Find the height of the flagpole. 37. A loading dock is 8 feet high. If a ramp makes an angle of 34 with the ground and is attached to the loading dock, how long is the ramp? 38. A loading dock is 5 feet high. If a ramp makes an angle of 25 with the ground and is attached to the loading dock, how long is the ramp? 41. Jeremy is flying his kite at Lake Michigan and lets out 200 feet of string. If his eyes are 6 feet from the ground as he sights the kite at an 55 angle of elevation, how high is the kite? (Assume that he is holding the kite near his eyes.) 42. Pam is flying a kite at the beach and lets out 120 feet of string. If her eyes are 5 feet from the ground as she sights the kite at a 64 angle of elevation, how high is the kite? (Assume that she is also holding the kite near her eyes.) 43. Doug is out hiking and walks to the edge of a cliff to enjoy the view. He looks down and sights a cougar at a 41 angle of depression. If the cliff is 60 meters high, approximately how far is the cougar from the base of the cliff? (Disregard Doug's height in your calculations.) 44. Juliette wakes up in the morning and looks out her window into the back yard. She sights a squirrel at a 54 angle of depression. If Juliette is looking out the window at a point which is 28 feet from the ground, how far is the squirrel from the base of the house? 45. From a point on the ground, the angle of elevation to the top of a mountain is 42 . Moving out a distance of 150 m (on a level plane) to another point on the ground, the angle of elevation is 33 . Find the height of the mountain. 46. From a point on the ground, the angle of elevation to the top of a mountain is 31 . Moving out a distance of 200 m (on a level plane) to another point on the ground, the angle of elevation is 18 . Find the height of the mountain. 39. Silas stands 600 feet from the base of the Sears Tower and sights the top of the tower. If the Sears Tower is 1,450 feet tall, approximate the angle of elevation from Silas' perspective as he sights the top of the tower. (Disregard Silas' height in your calculations.) 40. Mimi is sitting in her boat and sees a lighthouse which is 20 meters tall. If she sights the top of the lighthouse at a 32 angle of elevation, approximately how far is Mimi's boat from the base of the lighthouse? (Disregard Mimi's distance from the water in your calculations.) Math 1330, Precalculus The University of Houston Chapter 7: Trigonometric Applications to Triangles Exercise Set 7.2: Area of a Triangle Use the formula A = 12 bh to find the area of each of 7. the following triangles. (You may need to find the base and/or the height first, using trigonometric ratios or the Pythagorean Theorem.) Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. Note: Figures may not be drawn to scale. 1. 57o 9 in 8. 38o 8 cm 10 cm 7m 2. 7 in In ABC below, AD BC . Use the diagram below to answer the following questions. A 3 in b c h 3. 8 ft 3 ft C D B a 9. 4. (b) Using the equation from part (a), solve for h. 61 cm 6 cm 5. (a) Use ACD to write a trigonometric ratio that involves C , b, and h. 10 m (c) If a represents the base of ABC , and h represents the height, then the area K of ABC is K = 12 ah . Substitute the equation from part (b) into this equation to obtain a formula for the area of ABC that no longer contains h. 10. (a) Use ABD to write a trigonometric ratio that involves B , c, and h. 30o (b) Using the equation from part (a), solve for h. 6. 8m 45o Math 1330, Precalculus The University of Houston (c) If a represents the base of ABC , and h represents the height, then the area K of ABC is K = 12 ah . Substitute the equation from part (b) into this equation to obtain a formula for the area of ABC that no longer contains h. Chapter 7: Trigonometric Applications to Triangles Exercise Set 7.2: Area of a Triangle Find the area of each of the following triangles. Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. Note: Figures may not be drawn to scale. C 16. 50o 8 cm 4 cm B 11. 120o 6 in A 10 in A C 12. T J 17. 53o B 4.1 m 11 cm 36o 30o A 13. 5.6 m M C 10 cm A S 18. L 7.9 m 21o 9m 125o Q 45 J o 3.5 m K 8m 14. R F 5 ft 135o D 19. Find the area of an isosceles triangle with legs measuring 7 inches and base angles measuring 22.5 . E 7 ft 15. G 12 m 28o D 8m O Answer the following. Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. 20. Find the area of an isosceles triangle with legs measuring 10 cm and base angles measuring 75 . 21. Find the area of GHJ , where G = 120 , h = 8 cm , and j = 15 cm . 22. Find the area of PQR , where R = 47 , p = 7 in , and q = 10 in . Math 1330, Precalculus The University of Houston Chapter 7: Trigonometric Applications to Triangles Exercise Set 7.2: Area of a Triangle 23. Find the area of TUV , where T = 28 , U = 81 , t = 12 m , and u = 6.5 m . 24. Find the area of FUN , where F = 92 , N = 28 , f = 9 cm , and n = 8 cm . In the following questions, the radius of circle O is given, as well as the measure of central angle AOB. Find the area of the segment of circle O bounded by AB and AB . Give exact values whenever possible. Otherwise, round answers to the nearest hundredth. 25. (a) Find all solutions to sin ( x ) = 0.5 for O 0 < x < 180 . (b) If the area of RST is 20 cm2, r = 16 cm and t = 5 cm , find all possible measures for S . 26. (a) Find all solutions to sin ( x ) = 0.2 for A 37. Radius: 8 cm Central Angle: 0 < x < 180 . (b) If the area of PHL is 12 m2, p = 10 m and h = 6 m , find all possible measures for L . 2 27. If the area of TRG is 37 in , t = 8 in and g = 12.5 in , find all possible measures for R . 28. If the area of PSU is 20 2 cm2, p = 4 cm and s = 5 cm , find all possible measures for U . 29. A regular octagon is inscribed in a circle of radius 12 in. Find the area of the octagon. B 3 7 38. Radius: 7 in Central Angle: 4 5 39. Radius: 4 in Central Angle: 3 40. Radius: 6 cm Central Angle: 3 4 30. A regular hexagon is inscribed in a circle of radius 11 cm. Find the area of the hexagon. 31. A regular hexagon is circumscribed about a circle of radius 6 in. Find the area of the hexagon. 32. A regular hexagon is circumscribed about a circle of radius 10 cm. Find the area of the hexagon. 33. Find the area of an equilateral triangle with side length 4 ft. 34. Find the area of a regular hexagon with side length 10 cm. 35. Find the area of a regular decagon with side length 8 in. 36. Find the area of a regular octagon with side length 12 m. Math 1330, Precalculus The University of Houston Chapter 7: Trigonometric Applications to Triangles Math 1330 Precalculus Electronic Homework (EHW 9) Sections 7.1, 7.2, 7.3 Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in the Online Textbook under the Exercises link in the appropriate section. Record your answers by logging into your CASA account, go to the EMCF link in the top menu bar, selecting the appropriate assignment in the EMCF menu and recording your choices on the EMCF form. You MUST record your answers by the deadline given. No late homework will be accepted. This homework has 25 questions; your grade will be out of 25 points. Some questions on this homework require a calculator; you can use any scientific calculator on those. Exam questions will not require a calculator. Section 7.1 1. Problem 7.1.4 a A. sin 60 9 x tan 60 y 9 B. cos 60 9 x tan 60 y 9 C. tan 60 x 9 cos 60 y 9 D. cos 30 9 x tan 60 y 9 E. cos 60 9 x tan 60 9 y F. None of these 2. Problem 7.1.4 b A. x 18 y 9 3 B. x 18 y 3 3 C. x 6 3 y 9 3 D. x 9 3 y 18 Page 1 of 9 E. x 9 3 2 y 9 3 F. None of these 3. Problem 7.1.8 a A. sin 45 x 11 cos 45 y 11 B. cos 45 11 x sin 45 11 y C. cos 45 x 11 sin 45 y 11 D. cos 45 x 11 sin 45 11 y E. sin 45 11 x cos 45 11 y F. None of these 4. Problem 7.1.32 b A. 45 , 45 , 90 B. 45 . 6 , 45 . 6 , 88 . 8 C. 69 . 5 , 69 . 5 , 41 . 0 D. 20 . 5 , 20 . 5 , 139 . 0 E. 44 . 4 , 44 . 4 , 91 . 2 F. None of these 5. Problem 7.1.38 b A. 25.5 feet B. 5.5 feet C. 11.8 feet D. 10.7 feet Page 2 of 9 E. 21.1 feet F. None of these 6. Problem 7.1.40 b A. 32.0 meters B. 17.0 meters C. 12.5 meters D. 23.6 meters E. 37.7 meters F. None of these Section 7.2 7. Problem 7.2.12 A. B. 55 3 cm2 2 55 cm2 C. 55 cm2 2 D. 5 21 cm2 E. 110 cm2 F. None of these 8. Problem 7.2.14 A. 35 3 ft 2 4 B. 35 3 ft 2 2 Page 3 of 9 C. 35 2 ft 2 4 D. 35 2 ft 2 2 E. 35 2 ft 4 F. None of these 9. Problem 7.2.22 A. 23 . 87 in 2 B. 25 . 60 in 2 C. 51 . 19 in 2 D. 51 . 31 in 2 E. 25.20 in2 F. None of these 10. Problem 7.2.32 A. 100 3 cm 2 B. 150 3 cm 2 C. 200 3 c m 2 D. 400 3 c m 2 E. 200 cm2 F. None of the above 11. Problem 7.2.36 A. 59.65 m2 Page 4 of 9 B. 62.35 m2 C. 119.29 m2 D. 347.65 m2 E. 695.29 m2 F. None of these 12. Problem 7.2.38 (Round to the nearest tenth.) A. 370.4 in2 B. 81.4 in2 C. 47.2 in2 D. 32.8 in2 E. 320.8 in2 F. None of these Section 7.3 13. Problem 7.3.4 A. 4 2 mm B. 8 3 mm 3 C. 16 mm D. E. 2 mm 16 No such triangle exists. F. None of these 14. Problem 7.3.6 A. 51 . 32 B. 38 . 68 Page 5 of 9 C. 41 . 41 D. 67 . 98 E. No such triangle exists. F. None of these 15. Problem 7.3.12 A. 6.83 cm B. 7.32 cm C. 46.61 cm D. 53.52 cm E. No such triangle exists. F. None of these 16. Problem 7.3.16 A. 6 . 36 B. 37 . 15 C. 37 . 15 , 142 . 85 D. 37 . 15 , 137 . 15 E. No such triangle exists. F. None of these 17. Problem 7.3.18 A. 50 . 34 , 129 . 66 B. 50 . 34 C. 39 . 76 , 140 . 34 D. 39 . 76 E. No such triangle exists. F. None of these Page 6 of 9 18. Problem 7.3.22 A. 11 . 84 B. 34 . 75 , 145 . 25 C. 34 . 75 D. 36 . 86 E. There is no such triangle. F. None of these 19. Problem 7.3.30 (find the measure(s) of angle B) A. 26 . 63 B. 53 . 78 C. 80 . 41 D. 99 . 59 E. There is no such triangle. F. None of these 20. Problem 7.3.30 (find the measure(s) of angle A) A. 53 . 78 B. 26 . 63 C. 80.6 D. 99 . 59 E. There is no such triangle. F. None of these 21. Problem 7.3.8 A. 1.39 in B. 3.85 in Page 7 of 9 C. 1.62 in D. 4.50 in E. No such triangle exists. F. None of these 22. Problem 7.3.10 A. 14 3 mm 3 B. 7 6 mm C. 7 2 mm D. 14 6 mm E. No such triangle exists. F. None of these 23. Problem 7.3.14 A. 3.70 ft B. 6.49 ft C. 7.21 ft D. 7.84 ft E. No such triangle exists. F. None of these Page 8 of 9 24. Problem 7.3.16 A. 37 . 15 , 142 . 85 B. 37.15 C. 37 . 15 , 137 . 15 D. 6.36 E. No such triangle exists. F. None of these 25. Problem 7.3.20 A. 13 . 85 B. 68 . 87 C. 41 . 13 D. 41 . 13 , 68 . 87 E. There is no such triangle. F. None of these Page 9 of 9 Math 1330 Precalculus Electronic Homework (EHW 9) Sections 7.1, 7.2, 7.3 Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in the Online Textbook under the Exercises link in the appropriate section. Record your answers by logging into your CASA account, go to the EMCF link in the top menu bar, selecting the appropriate assignment in the EMCF menu and recording your choices on the EMCF form. You MUST record your answers by the deadline given. No late homework will be accepted. This homework has 25 questions; your grade will be out of 25 points. Some questions on this homework require a calculator; you can use any scientific calculator on those. Exam questions will not require a calculator. Section 7.1 1. Problem 7.1.4 a A. sin 60 9 x tan 60 y 9 B. cos 60 9 x tan 60 y 9 C. tan 60 x 9 cos 60 y 9 D. cos 30 9 x tan 60 y 9 E. cos 60 9 x tan 60 9 y F. None of these 2. Problem 7.1.4 b A. x 18 y 9 3 B. x 18 y 3 3 C. x 6 3 y 9 3 D. x 9 3 y 18 Page 1 of 9 E. x 9 3 2 y 9 3 F. None of these 3. Problem 7.1.8 a A. sin 45 x 11 cos 45 y 11 B. cos 45 11 x sin 45 11 y C. cos 45 x 11 sin 45 y 11 D. cos 45 x 11 sin 45 11 y E. sin 45 11 x cos 45 11 y F. None of these 4. Problem 7.1.32 b A. 45 , 45 , 90 B. 45 . 6 , 45 . 6 , 88 . 8 C. 69 . 5 , 69 . 5 , 41 . 0 D. 20 . 5 , 20 . 5 , 139 . 0 E. 44 . 4 , 44 . 4 , 91 . 2 F. None of these 5. Problem 7.1.38 b A. 25.5 feet B. 5.5 feet C. 11.8 feet D. 10.7 feet Page 2 of 9 E. 21.1 feet F. None of these 6. Problem 7.1.40 b A. 32.0 meters B. 17.0 meters C. 12.5 meters D. 23.6 meters E. 37.7 meters F. None of these Section 7.2 7. Problem 7.2.12 A. B. 55 3 cm2 2 55 cm2 C. 55 cm2 2 D. 5 21 cm2 E. 110 cm2 F. None of these 8. Problem 7.2.14 A. 35 3 ft 2 4 B. 35 3 ft 2 2 Page 3 of 9 C. 35 2 ft 2 4 D. 35 2 ft 2 2 E. 35 2 ft 4 F. None of these 9. Problem 7.2.22 A. 23 . 87 in 2 B. 25 . 60 in 2 C. 51 . 19 in 2 D. 51 . 31 in 2 E. 25.20 in2 F. None of these 10. Problem 7.2.32 A. 100 3 cm 2 B. 150 3 cm 2 C. 200 3 c m 2 D. 400 3 c m 2 E. 200 cm2 F. None of the above 11. Problem 7.2.36 A. 59.65 m2 Page 4 of 9 B. 62.35 m2 C. 119.29 m2 D. 347.65 m2 E. 695.29 m2 F. None of these 12. Problem 7.2.38 (Round to the nearest tenth.) A. 370.4 in2 B. 81.4 in2 C. 47.2 in2 D. 32.8 in2 E. 320.8 in2 F. None of these Section 7.3 13. Problem 7.3.4 A. 4 2 mm B. 8 3 mm 3 C. 16 mm D. E. 2 mm 16 No such triangle exists. F. None of these 14. Problem 7.3.6 A. 51 . 32 B. 38 . 68 Page 5 of 9 C. 41 . 41 D. 67 . 98 E. No such triangle exists. F. None of these 15. Problem 7.3.12 A. 6.83 cm B. 7.32 cm C. 46.61 cm D. 53.52 cm E. No such triangle exists. F. None of these 16. Problem 7.3.16 A. 6 . 36 B. 37 . 15 C. 37 . 15 , 142 . 85 D. 37 . 15 , 137 . 15 E. No such triangle exists. F. None of these 17. Problem 7.3.18 A. 50 . 34 , 129 . 66 B. 50 . 34 C. 39 . 76 , 140 . 34 D. 39 . 76 E. No such triangle exists. F. None of these Page 6 of 9 18. Problem 7.3.22 A. 11 . 84 B. 34 . 75 , 145 . 25 C. 34 . 75 D. 36 . 86 E. There is no such triangle. F. None of these 19. Problem 7.3.30 (find the measure(s) of angle B) A. 26 . 63 B. 53 . 78 C. 80 . 41 D. 99 . 59 E. There is no such triangle. F. None of these 20. Problem 7.3.30 (find the measure(s) of angle A) A. 53 . 78 B. 26 . 63 C. 80.6 D. 99 . 59 E. There is no such triangle. F. None of these 21. Problem 7.3.8 A. 1.39 in B. 3.85 in Page 7 of 9 C. 1.62 in D. 4.50 in E. No such triangle exists. F. None of these 22. Problem 7.3.10 A. 14 3 mm 3 B. 7 6 mm C. 7 2 mm D. 14 6 mm E. No such triangle exists. F. None of these 23. Problem 7.3.14 A. 3.70 ft B. 6.49 ft C. 7.21 ft D. 7.84 ft E. No such triangle exists. F. None of these Page 8 of 9 24. Problem 7.3.16 A. 37 . 15 , 142 . 85 B. 37.15 C. 37 . 15 , 137 . 15 D. 6.36 E. No such triangle exists. F. None of these 25. Problem 7.3.20 A. 13 . 85 B. 68 . 87 C. 41 . 13 D. 41 . 13 , 68 . 87 E. There is no such triangle. F. None of these Page 9 of 9