Precalculus Second Semester Final Review This packet will prepare you for your second semester final exam. You will find a formula sheet on the back page; these are the same formulas you will receive for your final exam. This packet, as well as the final, should be completed using a scientific calculator only. Chapter 4A Trigonometry 1. Refer to ABC. Find a. sin A b. sec C c. cos A d. tan C e. cot A f. csc C 2. Find DE and EF to the nearest hundredth. DE = EF = 3. A flagpole casts a shadow 50 feet long when the elevation of the sun is 25. How tall is the flag pole? 4. Convert to degrees without using a calculator. π 5 radians 5. Convert to exact radians without using a calculator. 135
Give the exact value. 6. cos π 3 7. sin 30 8. cos 240 9. sin 3π 4 10. tan 2π 3 11. tan π 4 12. Find all values of θ between 0 and 2π such that cos θ = 3 2. 13. Find two values of θ between 0 and 2π such that cos θ = 1 2. 14. If cos θ = 4 and 0 θ π, find sin θ. 5 2 Chapter 4B Graphs of Trig Functions Evaluate without a calculator. Give an exact answer in radians. 15. cos 1 ( 1 2 ) 16. arc cos ( 1 2 ) 17. sin 1 ( 18. tan 1 3 ( ) 2 3 19. tan 1 (1) 3 ) 20. Complete the following table: f(θ) = sin θ g(θ) = cos θ h(θ) = tan θ Domain Range Zeros Period
Give a) the period and b) the amplitude of each function. 21. y = sin x 3 a) b) 22. y = 4cos 2x a) b) Sketch one cycle of the graph without using a graphing device. 23. y = sin 2x 24. y = 4cos π 2 x 25. Multiple Choice. Which is the equation that corresponds to the following graph? (a) 2y = sin π 2 x (b) y = 2sin x (c) y = 2sin 1 4 x (d) y = 1 sin 4x 2 Match each equation with its graph below. (HINT: Solve for y 1 st!!) 26. y 2 = sin x 2 27. 2y = sin 2x 28. 2y = sin πx 2 29. 2y = sin x 2 30. y = sin 2x 2 31. y 2 = sin 2πx
32. Consider the graph of the function f(x) = sin(x π 2 ). What is its phase shift from the parent sine curve? Sketch a graph of the function. 33. y = cos(x π) Phase shift: 34. f(x) = cos (x π 2 ) 1 Phase shift: Vertical shifts: Write a function whose graph will have the given characteristics. 35. Parent y = sin x; phase shift π 6, periodπ, amplitude 1 2 36. Parent y = cos x; phase shift π, period π 3, amplitude 1 2 37. Parent y = cos x; phase shift π, period 2π, amplitude 4 3 38. Parent y = sin x; phase shift π, period π, amplitude 2 4 Sketch a graph of the function described. 39. y = 2cos(x + π) 1 Amplitude= Phase shift= Vertical Shift=
40. Give an equation for the sine wave. Chapter 6: Law of Sines & Law of Cosines 41. Consider DEF where DE=38, EF= 48, and DF= 70. Find the measure of the given angle to the nearest tenth of a degree. a. D b. F 42. Three lifeguard stands are positioned as shown in the diagram. They would like to have a buoyant line that would run from stand I to stand III for non-swimmers. Approximately how long would the line have to be? 43. Use Law of Sines to find h.
Consider ABC where m A = 24, m B = 99, and c = 3.1. 44. Find the lengths of sides a and b. 45. Find the area of ABC. Chapter 5: Trig Identities & Solving Trig Equations 46. Use the figure below to determine the exact value of the given functions. a) 47. Find the exact value of given that 48. Find the exact value of using a sum or difference formula.
49. Find the exact value of the given expressions. a. b. Find all solutions for each equation on the interval 50. 51. 52. 53. 54. 55.
Chapter 10: Conic Sections 4 56. What is an equation for the hyperbola with vertices and and asymptote y = 3 x? a. c. b. d. 57. Identify the conic given by 58. Identify the center, vertices, and foci. Then graph 59. Write an equation for an ellipse with foci and vertices. 60. Write an equation for an ellipse with foci and major axis of length 6 61. Graph and write the equation for the hyperbola with foci, and major axis of length 3.
62. Put in standard form. Identify the conic section and identify its major parts 63. Put in standard form. Identify the conic section and identify its major parts 64. The profit P (in hundreds of dollars) that a company makes depends on the amount x (in hundreds of dollars) the company spends on advertising according to the model. What expenditure for advertising will yield a maximum profit? Chapters 6 & 10: PolarGraphs 65. Convert the point in rectangular coordinates to polar coordinates. Give exact answers when possible. a. b. c)
66. Convert the point in polar coordinates to rectangular coordinates. Give exact answers when possible. a. b. c) 67. Convert the polar equations to rectangular equations: a) r = 4 b) 68. Convert the rectangular equations to polar equations: a) y = 3 b) 69. Match the point in polar coordinates with either A, B, C, or D on the graph.
Limits Find the limit of each function below. Use the Limit Theorem and show all work. 70. lim 2x3 3x2 +5x 1 x 4x 3 +2x 2 x 71. lim 4 3x 2x2 x x 3 +2x+5 (2x+1)(3x 2) 72. lim x 3x 5