Pythagorean Identity. Sum and Difference Identities. Double Angle Identities. Law of Sines. Law of Cosines

Transcription

1 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 sin 2 θ + cos 2 θ = 1 Sum and Difference Identities sin(u + v) = sin u cos v + cos u sin v sin(u v) = sin u cos v cos u sin v cos(u + v) = cos u cos v sin u sin v cos(u v) = cos u cos v + sin u sin v Double Angle Identities sin(2u) = 2 sin u cos u cos(2u) = cos 2 u sin 2 u 2 u cos(2u) = 1 2 sin cos(2u) = 2 cos 2 u 1 sin A a Law of Sines sin B sin C = = b c Law of Cosines a 2 = b 2 + c 2 2bc cos A b 2 = a 2 + c 2 2ac cos B c 2 = a 2 + b 2 2ab cos C 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.

2 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) 11π 6 (B) 8π 9 (C) 7π 9 (D) 11π 12 (E) None of these 2. Convert an angle measuring 13π 10 (A) 234 (B) 240 (C) 246 (D) 252 (E) None of these radians into degrees. 3. If angle θ is in standard position and θ = 445, the terminal side of θ would lie in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4

3 4. If angle θ is in standard position and θ = 535, the terminal side of θ would lie in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 5. If angle θ is in standard position and θ = 19π, the terminal side of θ would lie in which quadrant? 12 (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 6. If angle θ is in standard position and θ = 7, the terminal side of θ would lie in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant 4 7. If angle θ is in standard position and θ = 160, which one of the following is coterminal with θ? (A) 20 (B) 160 (C) 220 (D) 520 (E) 200

4 8. If angle θ is in standard position and θ = 11π, which one of the following is coterminal with θ? 8 (A) 3π 8 (B) 5π 8 (C) 5π 8 (D) 11π 8 (E) 19π 8 9. A circle of radius 4 inches is intercepted by central angle θ. If the intercepted arc length is 20 inches, determine θ. (A) θ = 5 radians (B) θ = 5 (C) θ = π 36 radians (D) θ = π A circle of radius 10 inches is intercepted by central angle θ. If θ = 40, determine the area of the intercepted sector. (A) 10π 9 in.2 (B) 200in. 2 (C) 100π in. 2 9 (D) 200π in. 2 9

5 11. The area of a sector of a circle is 20 in 2. If the radius of the circle is 8 inches, what is the length of the intercepted arc? (A) π 36 in. (B) 5in. (C) 10in. (D) π 18 in. 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 feet tall. (B) The building is 16 3 feet tall. (C) The building is 48 3 feet tall. (D) The building is feet tall. 13. The right triangle below has a height of 16 meters. Write a function P(θ) representing the perimeter of the triangle in terms of angle θ. (A) P(θ) = 16 + b + h 16 (B) P(θ) = tan θ + 16 sin θ θ (C) P(θ) = 16 + tan θ 16 + sin θ 16 (D) P(θ) = tan θ + 16 sin θ

6 14. If tan θ > 0 and csc θ < 0, then θ lies in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant If sec θ > 0 and sin θ < 0, then θ lies in which quadrant? (A) Quadrant 1 (B) Quadrant 2 (C) Quadrant 3 (D) Quadrant The terminal side of angle θ, in standard position, passes through the point (5, 12). Find the exact value of csc θ and sec θ. (A) csc θ = and sec θ = 12 5 (B) csc θ = and sec θ = 13 5 (C) csc θ = and sec θ = 5 13 (D) csc θ = and sec θ = Given csc θ = 4 and cot θ < 0, determine: sin 2 θ + cos 2 θ. (A) (B) (C) (D) 14 16

7 18. Use the complimentary angle theorem and fundamental identities to find the exact value of: (A) 1 (B) 2 (C) 0 (D) 1 sin 50 cos 40 + cos 50 sin Use the complimentary angle theorem and fundamental identities to find the exact value of: (A) 1 (B) 0 (C) 1 (D) 2 cot 2 (40 ) + 1 sec 2 (50 ). 20. Given that cos(36 ) =.81, determine the value of: sin(36 ) csc(36 ) + 1 csc(54 ). (A) 1 sin 9 (B) 1.09 (C) 1.81 (D) 1 + sin(54 ) 21. Find the exact value of sec ( 7π 4 ). (A) 2 (B) 2 (C) 2 (D) 2 (E) 2 2

8 22. Find the exact value of: tan ( 7π 2 ). (A) 0 (B) 1 (C) 1 (D) 2 (E) Undefined 23. Find the exact value of: sec 2 ( 7π 4 ) + 3 sin2 ( 10π 3 ). (A) 7 4 (B) 17 4 (C) 3 (D) Find the exact value of: sin ( 5π ) tan ( 5π ). 2 4 (A) 2 (B) 1 (C) 0 (D) 1 (E) Undefined 25. Given the function, y = 3 2 cos(4x 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

9 26. Given the function, y = 3 2 cos(4x 5), determine the period and phase shift. (A) Period: π 2, Phase shift: left 5 4 (B) Period: π 2, Phase shift: right 5 4 (C) Period: π 4, Phase shift: left 4 5 (D) Period: π, Phase shift: right Using the graph (right) determine the period of the function. The period is (A) 4 (B) 2π 3 (C) π 2 (D) π Using the same graph as #27, determine the value of A in the equation modeling the function y = A cos (B (x π )) + D. 6 (A) 6 (B) 6 (C) 4 (D) 3 (E) 3

10 29. Which of the following equations does NOT model the graph in #27? (A) y = 3 cos(3x.5π) 4 (B) y = 3 cos (3 (x + π 6 )) 4 (C) y = 3 sin(3x) 4 (D) y = 3 sin(3x) 4 (E) y = 3 sin (3 (x π 3 )) Which of the following equations does NOT represent the function graphed below? (A) y = 5 cos ( π (x 1)) 2 3 (B) y = 5 cos ( π 3 x + 2π 3 ) 2 (C) y = 5 sin ( π 3 (x 1 2 )) 2 (D) y = 5 sin ( π (x 2.5)) Assume the amplitude of a sinusoidal function is 5 and the period is 8. If f(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 f(7) = 2, maximum at f(11) = 7 (B) minimum at f ( 11 ) = 3, maximum at f(8) = 7 2 (C) minimum at f(7) = 3, maximum at f(11) = 7 (D) minimum at f(5.5) = 2, maximum at f(11) = 7

11 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. (A) h(t) = 0.4 cos ( 2π (x 1.3)) (B) h(t) = 0.6 cos ( 2π (x 1.3)) (C) h(t) = 0.4 cos ( 2π (x 1.3)) (D) h(t) = 0.6 cos ( 2π (x 1.9)) Find the exact value of: cos 1 ( 2 2 ). (A) π 4 (B) π 4 (C) 3π 4 (D) 5π 4 (E) 7π Find the exact value of: sin 1 ( 3 2 ). (A) π 3 (B) π 4 (C) π 6 (D) π 2

12 35. Find the exact value of: tan 1 ( 1 3 ). (A) 2π 3 (B) 5π 6 (C) π 3 (D) π 6 (E) 2π Find the exact value of: cos 1 (cos ( 4π 3 )). (A) 4π 3 (B) π 3 (C) π 3 (D) 2π Find the exact value of: sin 1 (tan(π)). (A) Undefined (B) 0 (C) π (D) 1 (E) π 2

13 38. Find the exact value of: sin 1 (cos ( 4π 3 )). (A) Undefined (B) 2π 3 (C) π 3 (D) π 6 (E) π Find the exact value of: cos(sin 1 (1)). (A) Undefined (B) 1 (C) 0 (D) 1 (E) π Find the exact value of: tan(sin 1 ( 1)). (A) Undefined (B) 1 (C) 0 (D) 1 (E) π 2

14 41. Find the exact value of: sin(tan 1 ( 1)). (A) 2 2 (B) 2 2 (C) π 4 (D) π 4 (E) 7π Find the exact value of: cos(tan 1 (2)). (A) Undefined (B) 1 5 (C) 2 3 (D) Find the exact value of: cot (cos 1 ( 3 2 )). (A) Undefined (B) 3 5 (C) 5 3 (D) 2 5 (E) 5 2

15 44. Express tan(cos 1 u) as an algebraic expression involving u. (A) 1 u2 u (B) 1+u2 u (C) (D) u 1 u 2 u 1+u 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? (A) θ = tan 1 ( ) (B) θ = tan 1 ( ) (C) θ = sin 1 ( ) (D) θ = sin 1 ( ) 46. Simplify the expression: tan θ sec θ csc θ. The result is (A) tan θ (B) cot θ (C) cot θ (D) tan θ 47. Simplify the expression: (sec θ 1)(sec θ + 1). The result is (A) cot 2 θ (B) tan 2 θ (C) tan 2 θ (D) cot 2 θ

16 48. Simplify the expression: cos θ(tan θ + cot θ). The result is (A) 1 (B) cos θ (C) sin θ (D) sec θ (E) csc θ 49. Simplify the expression: tan α+tan β cot α+cot β. The result is (A) sin2 (αβ) cos 2 (αβ) (B) 2 (C) tan α tan β (D) cot α cot β 50. Use the sum and difference identities to determine the exact value of: cos ( 17π 12 ). (A) (B) (C) (D) 2 6 4

17 51. Use the sum and difference identities to determine the exact value of: sin ( π 12 ). (A) (B) (C) (D) Find the exact value of the expression: sin ( π 4 ) cos ( π 12 ) + cos (π 4 ) sin ( π 12 ). (A) 1 2 (B) 2 2 (C) 3 2 (D) If α = tan 1 ( 4 3 ), determine the exact value of: sin (α + 3π 4 ). (A) 2 10 (B) (C) 2 10 (D) Use the sum and difference identities to simplify: cos (θ π 2 ). (A) sin θ

18 (B) cos θ (C) sin θ (D) cos θ

19 55. If β = tan 1 ( 2 ), determine the exact value of: sin(2β). 3 (A) (B) (C) (D) If the terminal side of angle θ, in standard position, passes through the point ( 5, 3), determine the exact value of: cos(2θ). (A) (B) (C) 1 (D) Solve the equation: 2 cos 2 x 1 = 0, for x on the interval from [0, 2π). (A) x = π 4, 7π 4 (B) x = π 4, 3π 4, 5π 4, 7π 4 (C) x = π 3, 2π 3, 4π 3, 5π 3 (D) No solution 58. Solve the equation: 2 sin(2x) + 1 = 0, for x on the interval from [0, 2π). (A) x = 2π 3, 4π 3 (B) x = 7π 6, 11π 6 (C) x = 7π, 11π, 19π, 23π (D) x = π 3, 2π 3, 4π 3, 5π 3

20 59. Solve the equation: 2 sin 2 x = sin x + 1, for x on the interval from [0, 2π). (A) x = 1 2, 1 (B) x = 1 2, 1 (C) x = 2π 3, π, 4π 3 (D) x = π 2, 7π 6, 11π Solve the equation: 4(1 + sin x) = cos 2 x, for x on the interval from [0, 2π). (A) x = 3π 2 (B) x = π 2 (C) x = 1 (D) x = π 61. Solve the equation: sin(2x) = cos x, for x on the interval from [0, 2π). (A) x = 0, 1 2 (B) x = π 6, π 2, 5π 6, 3π 2 (C) x = π 3, π 2, 3π 2, 5π 2 (D) x = 0, π, π, 5π 2 2 (E) No solution 62. Solve the equation: cos(2x) + 5 cos x + 3 = 0, for x on the interval from [0, 2π). (A) No solution (B) x = 2π 3, 4π 3 (C) x = 1 2, 2 (D) x = 7π 6, 11π 6

21 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 feet apart. (B) The runners are 300 feet apart. (C) The runners are feet apart. (D) The runners are 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 feel long. (C) The new ramp is 10 2 feet long. (D) The new ramp is 10 3 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) b = 10, B = 120, C = 125 (B) a = 2, c = 1, A = 120 (C) A = 50, b = 3, C = 85 (D) A = 100, B = 30, b = 6

22 67. Clint is building a swing set for his children. Each supporting end of the swing set is to be an A- frame constructed with two 10-foot-long 4 by 4 s 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 feet apart. (C) The footings should be feet apart. (D) The footings should be feet apart. 68. Determine the length of side c of the oblique triangle if a = 2, b = 3, C = 60. (A) c = (B) c = 2 2 (C) c = 7 (D) c =

23 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. y = sin x 70. y = cos x 71. y = tan x 72. y = csc x 73. y = sec x 74. y = cot x 75. y = sin 1 x 76. y = cos 1 x 77. y = tan 1 x (A) (B) (C) (D) (E) (F) (G) (H) (I)

24 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. 77. Sketch the graph of y = 5 sin (3x π ) 2 below. Be sure to graph at least one period and label 5 4 significant ordered pairs on your graph. 78. Sketch the graph of y = 2 cos ( π x π) + 4 below. Be sure to graph at least one period and label 3 5 significant ordered pairs on your graph.

25 79. Solve the following equations for θ on the interval [0, 2π). Show your work clearly and BOX your solution(s). a. cos(2θ) = 1 2 b. cos 2 θ sin 2 θ + sin θ = 0 c. sin 2 θ = 6(cos θ + 1) d. cos(2θ) + 6 sin 2 θ = 4 e. sin(2θ) sin θ = cos θ

26 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: d(t) = A cos(b(t C)) + D d(t) = 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: Phase Shift: Write the equation in the form: d(t) = A cos(b(t C)) + D d(t) =

27 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 16. D 36. D 37. B 57. B 58. C ANSWERS CONTINUE 17. B 38. E 59. D 18. A 39. C 60. A 19. B 40. A 61. B 20. C 41. B 62. B

28 d(t) = 17 cos ( 2π (t 3)) (many equations possible) a. θ = π 3, 2π 3, 4π 3, 5π 3 b. θ = 7π 6, 11π 6, π 2 c. θ = π d. θ = π 3, 2π 3, 4π 3, 5π 3 e. θ = π 2, 3π 2, π 4, 3π 4, 5π 4, 7π 4 d(t) = 260 cos ( 2π (t 18)) (many equations possible)