Name Date Trigonometry of the Right Triangle Class Work Unless otherwise directed, leave answers as reduced fractions or round to the nearest tenth. 1. Evaluate the sin, cos, and tan of θ(theta). 2. Evaluate the sin, cos, and tan of α (alpha). 3. Find x. Evaluate sin, cos, and tangent of β (beta) and γ (gamma). Round answers to the nearest hundredth. 4. Find x. 5. A right triangle has an acute angle of α, and cos α = 9. What is sin α? 6. A right triangle has a hypotenuse of 7 and an angle of 40, find the larger leg (in any triangle, the longest side is opposite the largest angle, the shortest side is opposite the smallest angle, and the middle side is opposite the middle angle). 7. A right triangle has an angle of 50 and a longer leg of 8, find the hypotenuse. 8. Solve right triangle ABC using the measurements provided and the diagram shown. a. B = 50, a = 10 c. A = 36, c = 20 b. A = 72, a = 12 d. B = 17, b = 24 41 9. Evaluate all six trig functions of the angle, β. Give answers as exact values. a. b. 10. Let θ be an acute angle in a right triangle. Find the values of the other 5 trig functions. Give answers as exact values in simplified form. a. sin θ = 8 9 b. cos θ = 3 4 c. csc θ = 10 3
Trigonometry of the Right Triangle Homework Round answers to the nearest tenth. 11. Evaluate the sin, cos, and tan of θ(theta). 12. Evaluate the sin, cos, and tan of α (alpha). 13. Find x. Evaluate sin, cos, and tangent of β (beta) and γ (gamma). Round answers to the nearest hundredth. 14. Find x. Evaluate tangent of 17. 15. A right triangle has an acute angle of α, and sin α = 9. What is tan α? 16. A right triangle has a hypotenuse of 9 and an angle of 60, find the larger leg. 17. A right triangle has an angle of 20 and a longer leg of 5, find the hypotenuse. 18. Solve right triangle ABC using the measurements provided and the diagram shown. a. B = 40, a = 19 c. A = 28, c = 11 b. A = 50, a = 3.1 d. B = 35, b = 24 41 19. Evaluate all six trig functions of the angle, β. Give answers as exact values in simplified form. a. b. 20. Let θ be an acute angle in a right triangle. Find the values of the other 5 trig functions. Give answers as exact values in simplified form. a. sin θ = 3 8 b. csc θ = 3 2 4 c. tan θ = 10 3 ~2~
Inverse Trig Functions Class Work 20. θ = 22. α = 23. β = 24. γ = 25. A right triangle has legs of 7 and 12. Find the smaller acute angle. 26. A right triangle has a longer leg of 5, and a hypotenuse of 13. Find the larger acute angle. ~3~
Inverse Trig Functions Homework 27. θ = 28. α = 29. β = 30. γ = 31. A right triangle has legs of 6 and 10. Find the larger acute angle. 32. A right triangle has a longer leg of 1.2, and a hypotenuse of 2. Find the larger acute angle. ~4~
Problem-Solving Classwork 33. A tree is 75 feet tall. What is the length of its shadow if the angle it makes with the sun is 55? 34. Sally is wearing 10-inch heels. IF the platform of her shoe is 5 inches, and her foot from heel to arch is 8 inches, what is the angle of elevation of her foot? 35. Tony spots a radio tower at the top of a building. He is standing 50 meters from the base of the building. If the angles of elevation of his sight are 58 and 62 to the bottom and the top of the tower respectively, what is the height of the tower? ~5~
Problem-Solving Homework 36. Snow White is being held prisoner in a tower with a window 40 feet from the ground. Prince Charming shows up with a 45-foot ladder. If the maximum angle that the ladder can make with the ground is 72, will the prince be able to rescue Snow White? Expain your answer. 37. A ski lift takes skiers 500 meters up from the base of a ski trail to the top. If the elevation at the top is 2200 meters and the elevation at the bottom is 1800 meters, what is the angle of elevation of the ski lift? 38. Two points in front of a building are 25 feet apart. The angles of elevation from the points to the top of the building are 18 and 25. How tall is the building? ~6~
Special Right Triangles Classwork 39. g = 40. m = 41. p = h = n = q = ~7~
Special Right Triangles Homework 42. g = 43. m = 44. p = h = n = q = ~8~
Law of Sines Class Work Solve triangle ABC. 45. A = 70, B = 30, c = 4. 46. B = 65, C = 50, a = 12 47. b = 6, A = 25, B = 45 48. c = 8, B = 60, C = 40 49. c = 12, b = 6, C = 70 50. b = 12, a = 15, B = 40 51. A = 35, a = 6, b = 11 52. A swimmer is swimming in the ocean and gets a leg cramp in between two lifeguard stands. If the stands are 320 feet apart, and the angles from the stands to the swimmer are 50 and 45, find the distance from each lifeguard to the swimmer. 53. You are creating a triangular patio. One side is 29 feet long and another is 32 feet. The angle opposite the 32 ft. side is 75. If you want to put a brick border around the patio, how many feet of border do you need? If each brick is 8 inches long, about how many bricks will you need? ~9~
Law of Sines Homework Solve triangle ABC. 54. A = 60, B = 40, c = 5 55. B = 75, C = 50, a = 14 56. b = 6, A = 35, B = 45 57. c = 8, B = 50, C = 40 58. c = 12, b = 8, C = 65 59. b = 12, a = 16, B = 50 60. A = 40, a = 5, b = 12 61. A triangular pane in a stained glass window has two angles of 35 and 85. If the side between these angles is 75 cm, what are the lengths of the other two sides? 62. A roller coaster has a hill that goes up at a 70 angle with the ground. It gets to the top and turns 30 before dropping 418 feet. What is the length of the upward climb? ~10~
Law of Cosines Class Work Solve triangle ABC. 63. a = 3, b = 4, c = 6 64. a = 5, b = 6, c = 7 65. a = 7, b = 6, c = 4 66. A = 100, b = 4, c = 5 67. B = 60, a = 5, c = 9 68. C = 40, a = 10, b = 12 69. The airline distance from Los Angeles, CA to Phoenix, AZ is 367 miles. The distance from Phoenix to Salt Lake City, UT is 504 miles and the distance from Salt Lake City to Los Angeles is 579 miles. What is the angle formed from LA to Phoenix to Salt Lake City? 70. Cal C takes his 2 dogs for a walk. He lets them off their leash in a field where Edison runs at 7 m/s and Einstein runs at 6 m/s. Cal determines the angle between the dogs is 20, how far are the dogs from each other in 8 seconds? 71. A golfer hits a ball 30 to the left of straight towards the hole. If the length of his shot is 156 yards, and the length from the tee to the hole is 186 yards, how far is the golfer s ball from the tee? ~11~
Law of Cosines Home Work Solve triangle ABC. 72. a = 4, b = 5, c = 8 73. a = 4, b = 10, c = 13 74. a = 11, b = 8, c = 6 75. A = 85, b = 3, c = 7 76. B = 70, a = 6, c = 12 77. C = 25, a = 14, b = 19 78. The airline distance from Rome, Italy to Paris, France is 697 miles. The distance from Paris to Berlin, Germany is 545 miles and the distance from Berlin to Rome is 734 miles. What is the angle formed from Berlin to Paris to Rome? 79. A student takes his 2 dogs for a walk. He lets them off their leash in a field where Edison runs at 10 m/s and Einstein runs at 8 m/s. The student determines the angle between the dogs is 25, how far are the dogs from each other in 5 seconds? 80. A golfer hits a ball 15 to the right of straight towards the hole. If the length of his shot is 89 yards, and the length from the tee to the hole is 160 yards, how far is the golfer s ball from the tee? ~12~
Multiple Choice 1. Evaluate tan α. a. 0.9 b. 0.45 c. 0.5 d. 2 Unit Review 2. The longer leg of a right triangle is 6 and the smallest angle is 20, what is the hypotenuse? a. 6.4 b. 5.6 c. 17.5 d. 14.7 3. Find the value of a. a. 18.8 b. 0.03 c. 15 d. 20 4. Find the value of β. a. 56.3 b. 41.8 c. 33.7 d. 48.2 5. Find the exact value of sin β in simplest form. a. 18.0 b. c. d. 2 13 13 10 325 3 325 65 6. Given ABC, with A = 35, a = 5, & c = 7, find B. a. 18.418 b. 53.418 c. 91.582 d. both a and b 7. Let θ be an acute angle in a right triangle. If sec θ = 7, what is sin θ? 6 a. b. c. d. 1 7 13 6 7 13 13 7 8. Given ABC, with A = 50, a = 6, & c = 8, find B. a. 1.021 b. 40 c. 128.979 d. no solution ~13~
9. Given ABC, with a = 9, b = 6, & c = 8, find B. a. 6.188 b. 42.6 c. 40.8 d. 78.6 Extended Response 1. A state park hires a surveyor to map out the park. a. A and B are on opposite sides of the lake, if the surveyor stands at point C and measures angle ACB= 50 and CA= 400 and CB= 350, how wide is the lake? b. At a river the surveyor picks to spots, X and Y, on the same bank of the river and a tree, C, on opposite bank. X= 80 and Y= 50 and XY=300, how wide is the river? (Remember distance is measured along perpendiculars.) c. The surveyor measured the angle to the top of a hill at the center of the park to be 32. She moved 200 closer and the angle to the top of the hill was 43. How tall was the hill? ~14~
2. A 15 ladder is rated to have no more than a 70 angle and no less than a 40 angle. a. What is maximum rated distance the base of the ladder can be placed from the wall? b. How high up a wall can the ladder reach and be within the acceptable use limits? c. At what base angle should the ladder be placed to reach 10 up the wall? ~15~
Answer Key 1. sin θ = 8 17, cos θ = 15 17, tan θ = 8 15 2. sin α = 4 5, cos α = 3 5, tan α = 4 3 3. x = 14.4, sin β =. 83, cos β =.56, tan β =. 67 sin γ =. 56, cos γ =.83, tan γ = 1.5 4. x = 14.3 5. 40 41 6. 5.4 7. 10.4 8. a. b = 11.9, c = 15.6, A = 40, C = 90 b. b = 3.9, c = 12.6, B = 18, C = 90 c. a = 11.8, b = 16.2, B = 54, C = 90 d. a = 78.5, c = 82.1, A = 73, C = 90 9. a. Sin β = 3 5, cos β = 4 5, tan β = 3 4 csc β = 5 3, sec β = 5 4, cot β = 4 3 b. Sin β =.7, cos β =.7, tan β = 1 csc β = 10 7 10, sec β =, cot β = 1 7 10. a. cos θ = 17 8 17, tan θ = 9 17 csc θ = 9 8 9 17 17, sec θ =, cot θ = 17 8 13. x 9.7, sin β =. 58, cos β =.81, tan β =. 72 sin γ =. 81, cos γ =.58, tan γ = 1.39 14. x = 7.7, cos 17 = 0.96 15. 9 40 16. 7.8 17. 5.3 18. a. b = 15.9, c = 24.8, A = 50, C = 90 b. b = 2.6, c = 4, B = 40, C = 90 c. a = 5.2, b = 9.7, B = 62, C = 90 d. a = 34.3, c = 41.8, A = 55, C = 90 19. a. Sin β = 2 5 csc β = 5 2 b. Sin β = 11 137 137 21 2 21, cos β =, tan β = 5 21 5 21 21, sec β =, cot β = 21 2 4 137 11, cos β =, tan β = 137 4 csc β = 137 137, sec β =, cot β = 4 11 4 11 20. a. cos θ = 55 3 55, tan θ = 8 55 csc θ = 8 3 8 55 55, sec θ =, cot θ = 55 3 b. Sin θ = 2 2, cos 1, tan θ = 2 2 3 3 b. Sin θ = 13 4, tan θ = 39 3 sec θ = 3, cot θ = 2 4 csc θ = 4 13 13 c. Sin θ = 3 10 4 3 39, sec θ =, cot θ = 3 13 91 3 91, cos θ =, tan θ = 10 91 sec θ = 10 91 91, cot θ = 91 3 11. sin θ = 5 13, cos θ = 12 13, tan θ = 5 12 12. sin α = 117 125, cos α = 44 125, tan α = 117 44 c. Sin θ = 10 109, cos θ = 3 109, 109 109 21. 28.1 22. 53.1 23. 56.3 24. 33.7 25. 30.3 csc θ = 109 109 sec θ =, cot θ = 3 10 3 10 ~16~
26. 22.6 27. 22.6 28. 69.4 29. 35.7 30. 54.3 31. 59 32. 36.9 33. 107.1 ft 34. 32 35. 14 m 36. Yes, the ladder can reach a maximum height of 43 feet. 37. 53.1 38. 27 feet (hint: set up and solve a system of equations) 39. g = 6, h = 6 3 40. m = 4.5 3, n = 4.5 41. p = 5 2, q = 5 2 42. g = 2 3, h = 2 3 3 43. m = 16 3 8 3, n = 3 3 44. p = 15, q = 15 2 45. C = 80, a = 3.82, b = 2.03 46. A = 65, b = 12, c = 10.14 47. C = 110, a = 3.59, c = 7.97 48. A = 80, a =12.26, b = 10.78 49. A = 82, B = 28, a = 12.65 50. A = 53.5, C = 86.5, c = 18.63 and A = 126.5, C = 13.5, c = 4.35 51. No Solution 52. 227.1 feet, 246.1 feet 53. 84 feet, about 126 bricks 54. C = 80, a = 4.40, b = 3.26 55. A = 55, b = 16.51, c = 13.09 56. C = 100, a = 4.87, c = 8.36 57. A = 90, a = 12.45, c = 9.53 58. A = 77.83, B = 37.17, a = 12.94 59. No Solution 60. No Solution 61. 438.1 ft 62. 49.7 cm and 86.3 cm 63. A = 26.384, B = 36.336, C = 117.280 64. A = 44.415, B = 57.122, C = 78.463 65. A = 86.417, B = 58.811, C = 34.772 66. a = 6.924, B = 34.670, C = 45.330 67. A = 33.668, b = 7.810, C = 86.332 68. A = 55.978, B = 84.022, c = 7.756 69. 81.7 70. 19.7 m 71. 93.1 yd 72. A = 24.147, B = 30.754, C = 125.1 73. A = 13.325, B = 35.184, C = 131.491 74. A = 102.636, B = 45.207, C = 32.157 75. a = 7.372, B = 23.916, C = 71.074 76. b = 11.435, A = 29.543, C = 80.453 77. c = 8.651, A = 43.149, B = 111.852 78. 71.3 79. 21.8 m 80. 77.5 yd MC1. C MC2. A MC3. A ~17~
MC4. B MC5. B MC6. C MC7. D MC8. D MC9. C ER1A. 320.2 ER1B. about 295 (hint: use 2 right triangles, set up and solve a system of 2 equations) ER1C. about 385 ER2A. 11.5 ER2B. 14.1 ER2C. 41.8 ~18~