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Review for Math 111 Final Exam The final exam is worth 30% (150/500 points). It consists of 26 multiple choice questions, 4 graph matching questions, and 4 short answer questions. Partial credit will be awarded on the short answer questions for quality of work shown. Calculators are not allowed. The following formula sheet will be given on the final exam. Trigonometric Formulas and Identities Pythagorean Identity sin2 π + cos 2 π = 1 Sum and Difference Identities sin(π’ + π£) = sin π’ cos π£ + cos π’ sin π£ sin(π’ β π£) = sin π’ cos π£ β cos π’ sin π£ cos(π’ + π£) = cos π’ cos π£ β sin π’ sin π£ cos(π’ β π£) = cos π’ cos π£ + sin π’ sin π£ Double Angle Identities sin(2π’) = 2 sin π’ cos π’ cos(2u) = cos 2π’ β sin 2π’ cos(2π’) = 1 β 2 sin 2π’ cos(2π’) = 2 cos 2π’ β 1 Law of Sines sin π΄ sin π΅ sin πΆ = = π π π Law of Cosines π2 = π 2 + π 2 β 2ππ cos π΄ π 2 = π2 + π 2 β 2ππ cos π΅ π 2 = π2 + π 2 β 2ππ cos πΆ You should know other formulas such as: arc length of a circle, area of a sector of a circle, Pythagorean identities, and Complimentary Angle Theorem. Part 1 Multiple Choice (4 points each) Each question has one correct answer. Partial credit is NOT possible. 1. Convert and angle measuring 165° into radians. (A) (B) (C) (D) 11π 6 8π 9 7π 9 11π 12 (E) None of these 2. Convert an angle measuring 13π 10 radians into degrees. (A) 234° (B) 240° (C) 246° (D) 252° (E) None of these 3. If angle π is in standard position and π = 445° , the terminal side of π would lie in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 4. If angle π is in standard position and π = β535° , the terminal side of π would lie in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 5. If angle π is in standard position and π = 19π 12 , the terminal side of π would lie in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 6. If angle π is in standard position and π = 7, the terminal side of π would lie in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 7. If angle π is in standard position and π = β160° , which one of the following is coterminal with π? (A) 20° (B) 160° (C) 220° (D) β520° (E) β200° 8. If angle π is in standard position and π = (A) (B) 11π 8 , which one of the following is coterminal with π? 3π 8 5π 8 (C) β 5π (D) β 11π (E) β 19π 8 8 8 9. A circle of radius 4 inches is intercepted by central angle π. If the intercepted arc length is 20 inches, determine π. (A) π = 5 πππππππ (B) π = 5° π (C) π = 36 πππππππ π° (D) π = 36 10. A circle of radius 10 inches is intercepted by central angle π. If π = 40° , determine the area of the intercepted sector. (A) 10π 9 ππ.2 (B) 200ππ.2 (C) (D) 100π 9 200π 9 ππ.2 ππ.2 11. The area of a sector of a circle is 20 ππ2 . If the radius of the circle is 8 inches, what is the length of the intercepted arc? (A) π 36 ππ. (B) 5ππ. (C) 10ππ. (D) π 18 ππ. 12. When a 6 foot tall person is standing 48 feet from the base of a flagpole, the angle of elevation to the top of the flagpole is 30° . What is the height of the flagpole? (A) The building is 16β3 + 6 feet tall. (B) The building is 16β3 feet tall. (C) The building is 48β3 feet tall. (D) The building is 48β3 + 6 feet tall. 13. The right triangle below has a height of 16 meters. Write a function π(π) representing the perimeter of the triangle in terms of angle π. 16 (A) π(π) = 16 + π + β (B) π(π) = 16 + 16 tan π + 16 sin π (C) π(π) = 16 + tan π 16 16 + sin π 16 16 (D) π(π) = 16 + tan π + sin π π 14. If tan π > 0 and csc π < 0, then π lies in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 15. If sec π > 0 and sin π < 0, then π lies in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 16. The terminal side of angle π, in standard position, passes through the point (5, β12). Find the exact value of csc π and sec π. (A) csc π = 12 and sec π = 13 13 (B) csc π = 13 and sec π = 12 13 13 5 5 5 (C) csc π = 12 and sec π = 13 13 (D) csc π = β 12 and sec π = 13 5 17. Given csc π = 4 and cot π < 0, determine: sin2 π + cos 2 π. 15 (A) β 16 16 (B) 16 14 (C) 16 14 (D) β 16 18. Use the complimentary angle theorem and fundamental identities to find the exact value of: sin 50° cos 40° + cos 50° sin 40° . (A) 1 (B) 2 (C) 0 (D) β1 19. Use the complimentary angle theorem and fundamental identities to find the exact value of: cot 2 (40° ) + 1 β sec 2 (50° ) . (A) β1 (B) 0 (C) 1 (D) 2 1 20. Given that cos(36° ) = .81, determine the value of: sin(36° ) β csc(36° ) + csc(54° ). (A) 1 β sin 9° (B) 1.09 (C) 1.81 (D) 1 + sin(54° ) 7π 21. Find the exact value of sec ( 4 ) . (A) β2 (B) 2 (C) β2 (D) ββ2 (E) β2 2 7π 22. Find the exact value of: tan ( 2 ) . (A) 0 (B) 1 (C) β1 (D) β2 (E) Undefined 7π 23. Find the exact value of: sec 2 ( 4 ) + 3 sin2 (β 10π 3 ). 7 (A) β 4 (B) 17 4 (C) 3 (D) 25 4 24. Find the exact value of: sin (β 5π 2 ) β tan (β 5π 4 ). (A) β2 (B) β1 (C) 0 (D) 1 (E) Undefined 25. Given the function, π¦ = 3 β 2 cos(4π₯ β 5), determine the amplitude and vertical shift. (A) Amplitude: 2, Vertical shift: 3 (B) Amplitude: -2, Vertical shift: 3 (C) Amplitude: 3, Vertical shift: -2 (D) Amplitude: -3, Vertical shift: 2 26. Given the function, π¦ = 3 β 2 cos(4π₯ β 5), determine the period and phase shift. π 5 (A) Period: 2 , Phase shift: left 4 π 5 (B) Period: 2 , Phase shift: right 4 π 4 (C) Period: 4 , Phase shift: left 5 π 4 (D) Period: 4 , Phase shift: right 5 27. Using the graph (right) determine the period of the function. The period is (A) β4 (B) (C) 2π 3 π 2 π (D) β 6 28. Using the same graph as #27, determine the value of A in the equation modeling the function π π¦ = π΄ cos (π΅ (π₯ β )) + π·. 6 (A) 6 (B) β6 (C) β4 (D) 3 (E) β3 29. Which of the following equations does NOT model the graph in #27? (A) π¦ = β3 cos(3π₯ β .5π) β 4 π (B) π¦ = 3 cos (3 (π₯ + 6 )) β 4 (C) π¦ = β3 sin(3π₯) β 4 (D) π¦ = 3 sin(3π₯) β 4 π (E) π¦ = 3 sin (3 (π₯ β 3 )) β 4 30. Which of the following equations does NOT represent the function graphed below? π (A) π¦ = 5 cos (3 (π₯ β 1)) β 2 π (B) π¦ = β5 cos ( 3 π₯ + π 2π 3 )β2 1 (C) π¦ = 5 sin ( 3 (π₯ β 2)) β 2 π (D) π¦ = β5 sin (3 (π₯ β 2.5)) β 2 31. Assume the amplitude of a sinusoidal function is 5 and the period is 8. If π(3) = 7 is a maximum value of the function, then where would a minimum value occur? Where would another maximum value occur? (A) minimum at π(7) = 2, maximum at π(11) = 7 11 (B) minimum at π ( ) = β3, maximum at π(8) = 7 2 (C) minimum at π(7) = β3, maximum at π(11) = 7 (D) minimum at π(5.5) = 2, maximum at π(11) = 7 32. A weight, attached to the end of a very long spring, is bouncing up and down. For a small period of time, this motion can be modeled by a sinusoidal function. When your stopwatch reads 1.3 seconds, the weight is at a minimum height of 2.4 feet above the floor. When your stop watch reads 1.9 seconds, the weight reaches the next maximum height of 3.2 feet. Determine the equation modeling the height of the weight, h, in terms of time, t. 2π (A) β(π‘) = β0.4 cos (1.2 (π₯ β 1.3)) + 2.8 2π (B) β(π‘) = β0.6 cos (1.2 (π₯ β 1.3)) + 2.4 2π (C) β(π‘) = β0.4 cos (0.6 (π₯ β 1.3)) + 2.4 2π (D) β(π‘) = β0.6 cos (0.6 (π₯ β 1.9)) + 2.8 33. Find the exact value of: cos β1 (β β2 ). 2 π (A) 4 π (B) β 4 (C) (D) (E) 3π 4 5π 4 7π 4 β3 34. Find the exact value of: sinβ1 ( 2 ). π (A) 3 π (B) 4 π (C) 6 π (D) 2 35. Find the exact value of: tanβ1 (β (A) (B) 1 ). β3 2π 3 5π 6 π (C) β 3 π (D) β 6 (E) β 2π 3 4π 36. Find the exact value of: cos β1 (cos ( 3 )). (A) 4π 3 π (B) β 3 (C) (D) π 3 2π 3 37. Find the exact value of: sinβ1(tan(π)). (A) Undefined (B) 0 (C) π (D) β1 π (E) 2 4π 38. Find the exact value of: sinβ1 (cos ( 3 )). (A) Undefined (B) 2π 3 π (C) β 3 π (D) 6 π (E) β 6 39. Find the exact value of: cos(sinβ1(1)). (A) Undefined (B) 1 (C) 0 (D) β1 π (E) 2 40. Find the exact value of: tan(sinβ1(β1)). (A) Undefined (B) 1 (C) 0 (D) β1 π (E) β 2 41. Find the exact value of: sin(tanβ1(β1)). (A) β2 2 (B) β β2 2 π (C) 4 π (D) β 4 (E) 7π 4 42. Find the exact value of: cos(tanβ1(2)). (A) Undefined (B) (C) (D) 1 β5 2 β3 2 β5 3 43. Find the exact value of: cot (cosβ1 (β 2)). (A) Undefined (B) β (C) β (D) β (E) β 3 β5 β5 3 2 β5 β5 2 44. Express tan(cos β1 π’) as an algebraic expression involving u. (A) (B) β1βπ’2 π’ β1+π’2 π’ π’ (C) β1βπ’2 π’ (D) β1+π’2 45. The length of the shadow of a building 34 meters tall is 37 meters. Which of the following would give the angle of elevation of the sun? 37 (A) π = tanβ1 (34) 34 (B) π = tanβ1 (37) 34 (C) π = sinβ1 (37) 37 (D) π = sinβ1 (34) 46. Simplify the expression: tan π β sec π csc π. The result is (A) β tan π (B) cot π (C) β cot π (D) tan π 47. Simplify the expression: (sec π β 1)(sec π + 1). The result is (A) cot 2 π (B) tan2 π (C) β tan2 π (D) β cot 2 π 48. Simplify the expression: cos π(tan π + cot π). The result is (A) 1 (B) cos π (C) sin π (D) sec π (E) csc π 49. Simplify the expression: tan πΌ+tan π½ cot πΌ+cot π½ . The result is sin2 (πΌπ½) (A) cos2(πΌπ½) (B) 2 (C) tan πΌ tan π½ (D) cot πΌ cot π½ 17π 50. Use the sum and difference identities to determine the exact value of: cos ( 12 ). (A) β6+β2 4 (B) β6ββ2 4 (C) (D) ββ6ββ2 4 β2ββ6 4 π 51. Use the sum and difference identities to determine the exact value of: sin (12). (A) β6+β2 4 (B) β6ββ2 4 (C) (D) ββ6ββ2 4 β2ββ6 4 π π π π 52. Find the exact value of the expression: sin ( 4 ) cos (12) + cos ( 4 ) sin (12). (A) 1 2 (B) β2 2 (C) β3 2 (D) 1 4 53. If πΌ = tanβ1 (β 3), determine the exact value of: sin (πΌ + 3π 4 ). β2 (A) β 10 (B) 7β2 10 β2 (C) 10 (D) β 7β2 10 π 54. Use the sum and difference identities to simplify: cos (π β 2 ). (A) sin π (B) cos π (C) β sin π (D) β cos π 2 55. If π½ = tanβ1 (β 3), determine the exact value of: sin(2π½). 12 (A) 13 (B) 12 β13 12 (C) β 13 (D) β 12 β13 56. If the terminal side of angle π, in standard position, passes through the point (β5, 3), determine the exact value of: cos(2π). (A) (B) 16 β34 19 34 (C) 1 (D) 8 17 57. Solve the equation: 2 cos2 π₯ β 1 = 0, for x on the interval from [0, 2π). π 7π (A) π₯ = 4 , 4 π 3π 5π 7π (B) π₯ = 4 , 4 , 4 , 4 π 2π 4π 5π (C) π₯ = 3 , 3 , 3 , 3 (D) No solution 58. Solve the equation: 2 sin(2π₯) + 1 = 0, for x on the interval from [0, 2π). (A) π₯ = 2π 4π (B) π₯ = 7π 11π (C) π₯ = 7π 11π 19π 23π 3 6 12 , 3 , , 6 12 , 12 , π 2π 4π 5π (D) π₯ = 3 , 3 , 3 , 3 12 59. Solve the equation: 2 sin2 π₯ = sin π₯ + 1, for x on the interval from [0, 2π). 1 (A) π₯ = β 2 , 1 1 (B) π₯ = 2 , β1 (C) π₯ = 2π 3 4π , π, 3 π 7π 11π (D) π₯ = 2 , , 6 6 60. Solve the equation: 4(1 + sin π₯) = cos2 π₯, for x on the interval from [0, 2π). (A) π₯ = 3π 2 π (B) π₯ = β 2 (C) π₯ = β1 (D) π₯ = π 61. Solve the equation: sin(2π₯) = cos π₯, for x on the interval from [0, 2π). 1 (A) π₯ = 0, 2 π π 5π 3π (B) π₯ = 6 , 2 , 6 , 2 π π 3π 5π (C) π₯ = 3 , 2 , 2 π (D) π₯ = 0, 2 , π, , 2 5π 2 (E) No solution 62. Solve the equation: cos(2π₯) + 5 cos π₯ + 3 = 0, for x on the interval from [0, 2π). (A) No solution (B) π₯ = 2π 4π 3 , 3 1 (C) π₯ = β 2 , β2 (D) π₯ = 7π 11π 6 , 6 63. Two runners, approaching the finish line, in a marathon determine that the angles of elevation of a news helicopter covering the race are 45° and 60° . If the helicopter is 300 feet directly above the finish line, how far apart are the runners? (A) The runners are 300 β 100β3 feet apart. (B) The runners are 300 feet apart. (C) The runners are 100β3 feet apart. (D) The runners are 300β3 feet apart. 64. A loading ramp 10 feet long that makes an angle of 45° with the horizontal is to be replaced by one that makes an angle of 30° with the horizontal. How long is the new ramp? (A) The new ramp is 20 feet long. (B) The new ramp is 10β6 3 feel long. (C) The new ramp is 10β2 feet long. (D) The new ramp is 10β3 feet long. 65. If a triangle does NOT have any right angles, then the triangle is called (A) obtuse (B) scalene (C) oblique (D) acute 66. Which of the following might result in two possible triangles? (A) π = 10, π΅ = 120° , πΆ = 125° (B) π = 2, π = 1, π΄ = 120° (C) π΄ = 50° , π = 3, πΆ = 85° (D) π΄ = 100° , π΅ = 30° , π = 6 67. Clint is building a swing set for his children. Each supporting end of the swing set is to be an Aframe constructed with two 10-foot-long 4 by 4βs joined at a 45° angle. To prevent the swing set from tipping over, Clint wants to secure the base of each A-frame in concrete footings. How far apart should the footings for each A-frame be? (A) The footings should be 10 feet apart. (B) The footings should be 200 β 100β3 feet apart. (C) The footings should be 20 β 10β2 feet apart. (D) The footings should be β200 β 100β2 feet apart. 68. Determine the length of side c of the oblique triangle if π = 2, π = 3, πΆ = 60° . (A) π = β13 β 6β3 (B) π = 2β2 (C) π = β7 (D) π = 13 β 6β2 Part 2: Match the trigonometric function with its correct graph below. Write the appropriate letter in the space provided. Select each letter at most once. 69. π¦ = sin π₯ ________ 72. π¦ = csc π₯ ________ 75. π¦ = sinβ1 π₯ ______ 70. π¦ = cos π₯ ________ 73. π¦ = sec π₯ ________ 76. π¦ = cos β1 π₯ ______ 71. π¦ = tan π₯ ________ 74. y = cot π₯ ________ 77. π¦ = tanβ1 π₯ ______ (A) (B) (C) (D) (E) (F) (G) (H) (I) Part 3 Short Answer Partial credit is possible on these short answer exercises. Show your work for full credit. Answers given without clear supporting work or reasonable explanation may receive little or no credit. π 77. Sketch the graph of π¦ = 5 sin (3π₯ β 4 ) β 2 below. Be sure to graph at least one period and label 5 significant ordered pairs on your graph. π 78. Sketch the graph of π¦ = β2 cos ( 3 π₯ β π) + 4 below. Be sure to graph at least one period and label 5 significant ordered pairs on your graph. 79. Solve the following equations for π on the interval [0, 2π). Show your work clearly and BOX your solution(s). 1 a. cos(2π) = β 2 b. cos 2π β sin 2π + sin π = 0 c. sin 2π = 6(cos π + 1) d. cos(2π) + 6 sin 2π = 4 e. sin(2π) sin π = cos π 80. A sinusoidal function has a maximum at the point (3, 38) and the next minimum at the point (21,4). The next maximum would occur at the point (______, ______). Graph two complete periods of this function. Vertical Shift: Amplitude: Period: Phase Shift: Write the equation in the form: π(π‘) = π΄ cos(π΅(π‘ β πΆ)) + π· π(π‘) = __________________________________________ 81. The Ferris Wheel: The tallest Ferris wheel in the world is the High Roller located on the Las Vegas strip. You decide to take a ride. 18 minutes into your ride, you reach the highest point at 550 feet above the ground. 33 minutes into your ride, you are at the lowest point 30 feet above the ground. Graph two complete periods of your ride showing distance above the ground against time. Vertical Shift: Amplitude: Period: Write the equation in the form: π(π‘) = π΄ cos(π΅(π‘ β πΆ)) + π· π(π‘) =_________________________________________ Phase Shift: Answers 21. C 42. B 63. A 1. D 22. E 43. A 64. C 2. A 23. B 44. A 65. C 3. A 24. C 45. B 66. B 4. C 25. A 46. B 67. D 5. D 26. B 47. B 68. C 6. A 27. B 48. E 69. E 7. C 28. E 49. C 70. A 8. C 29. D 50. D 71. B 9. A 30. C 51. B 72. F 10. C 31. C 52. C 73. I 11. B 32. A 53. B 74. D 12. A 33. C 54. A 75. H 13. D 34. A 55. C 76. C 14. C 35. D 56. D 77. G 15. D 36. D 57. B 16. D 37. B 58. C ANSWERS CONTINUEβ¦ 17. B 38. E 59. D 18. A 39. C 60. A 19. B 40. A 61. B 20. C 41. B 62. B 78. 81. 79. π(π‘) = 17 cos ( 2π (π‘ β 3)) + 21 36 (many equations possible) 82. 80. π 2π 4π 5π a. π = 3 , b. π = , , 6 ,2 3 3 3 7π 11π π 6 , c. π = π π 2π 4π 5π d. π = 3 , 3 , 3 , 3 π 3π π 3π 5π 7π e. π = , 2 2 , , 4 4 , 4 , 4 2π π(π‘) = 260 cos ( (π‘ β 18)) + 290 30 (many equations possible)