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PRECALCULUS GT/H REVIEW FOR FIRST SEMESTER FINAL Work the following on notebook paper. 1. Find all of the zeros of f x 6 x 4 x3 32 x 2 5x 10. 2. Find all of the zeros of f x 21x 4 95x3 109 x 2 105x 200. 3. Find two positive numbers such that their product is 192 and the sum of the first plus three times the second is a minimum. 4. A manufacturer wants to design an open box having a square base and a surface area of 108 sq. in. What dimensions produce a box with maximum volume? 5. Four feet of wire is to be used to form a square and a circle. Find the length of the side of the square and the radius of the circle so that the enclosed total area is: (a) a maximum (b) a minimum 6. Find all of the asymptotes of: x2 9 2 x 2 3x 2 (a) y 2 (b) y x 1 x 16 ________________________________________________________________________________ Solve. 7. x 6 x 14 8. x 1 1 x 6 9. x3 4x2 x 4 x2 x 6 10. 0 x 1 11. x 2 3x 1 12.Find two angles, one positive and one negative, that are coterminal with the given angle. 5 a. 35 b. 6 13. a. Convert 165 to radians. Give answers in terms of . b. Convert 208 to radians. Give the answer to the nearest hundredth of a radian. 14. a. Convert 7 to degrees. 6 b. Convert 1.8 radians to nearest tenth of a degree. 15. Write an equation for a sine wave that has amplitude 3 and period 8. 2 16. If is a third quadrant angle and cos , find sin . 3 17. Express each of the following in terms of a reference angle. a. cos236 b. sin485 c. sin( 62 ) 18. Give the exact value of the six trigonometric functions for each angle. a. b. 225 c. 6 2 Let cot x 19. 5 3 , where x . Find the values of the other five trigonometric functions. 12 2 20. Without using your calculator or a table, find the value of each expression. Leave your answers in terms of whenever appropriate. 3 2 a. Tan 11 b. Sin 1 e. Sin 1 0 f. sin(Cos 1 ) 3 ) 2 5 g. sec(Tan 1 ) 12 d. Tan 1 3 c. Cos 1 ( 3 5 h. cos( Sin 11) 21. Solve the equation to the nearest hundredth of a radian. 2 sec x 5 0 for 0 x 2 . 22. Evaluate without using a table or calculator. 5 a. sin120 b. sec 4 23. Evaluate without using tables or calculators. F 3 I b. cos( Arc tan G H2 J K a. Cos 1 3) 3 I F G H2 J K c. cos 3 1I F ) d. Arcsin(sin ) G J H2 K 4 c. sin( Sin 1 7 I F G H6 J K d. tan 1I F G H2 J K) e. csc(Tan 1 24. If sec x 5 and x 2 , find exact value for the other five trigonometric functions of x. 25. Determine the sinusoidal function of sine with amplitude 1 3 , period , and translation 2 units up. 26. Determine the angle coterminal with 27. A 4 9 7 2 that has a radian measure between 0 and 2 . clockwise rotation would terminate in which quadrant and yield what angle measurement? 28. Express the function sin 8 in terms of its cofunction. 29. Find the value of cos if sin 2 3 and angle is in standard position with terminal side in the fourth quadrant. 30. Sec and cot have opposite signs in which quadrant? 31. Determine the following: sin( Cos 1 5 ). 6 1 cos tan 2 sec 2 2 sin 32. Simplify the following . 33. Solve the equation: cos x sin x sin2 x for 0 x 2 . 34. Match the function to the graph. A. y 2 sin 2 x B. y 3cos 3x 1 2 5 2 E. y sin x F. y cos 3 2 C. y cos x x 3 2x H. y cos 3 x 2 D. y 2 sin G. y 2sin x 2 3 3 3 3 2 2 2 2 1 1 1 1 0 1 2 3 4 5 6 1 1 1 1 2 2 2 2 3 3 3 3 A. __________ B. _________ C. _________ D. _________ 3 3 3 3 2 2 2 2 1 1 1 1 0 1 2 3 4 5 6 1 1 1 1 2 2 2 2 3 3 3 3 E. __________ F. _________ 35. Find the exact value of sin 15. 36. Find the exact value of cos 3 8 . G. _________ H. _________ 37. Find sin 2 . 2 7 cos , given that 38. Graph with the correct trig function. A. y 2 3cos x C. B. y 2 cos x 3 y 3sin x 2 D. y 2 3sin x 39. A right triangle has an acute angle such that cot 15. Find cos . 40. Simplify: 1 1 . sec x 1 sec x 1 41. Evaluate: sec( Arc tan 3) . 42. Evaluate: FI. HK Arccos 1 2 43. Find the reference angle for 305 . 44. Find all solutions (exact value) in the interval 0, 2 , sec 3x 2 . 45. What is 1 tan 2 ? 46. What is the reference angle for 188 ? 47. Graph y 2 sec(2 x) 1. 48. Find the exact value of cos 7 . 12 49. Express 315 degrees in radians. 50. Express 8420'40" in decimal degrees. 51 Express in degrees, minutes, and seconds : 38.405 52. Find the value of cos of the angle in standard position that passes through point (3, 4). 53. An angle with rotation of – 220 degrees terminates in which quadrant? 54. Find the domain of the function : g( x) x 4 . 55. Graph y = 2tan x +1. f ( x) x 2 , g ( x) 3x 4 , evaluate 56. Given: y csc 57. Graph 1 x . bf ggbg x 1 2 58. Simplify the following. A. csc x cos x cot x B. tan2 x sin2 x cos2 x C. 1 1 tan x 2 1 1 cot 2 x 59. Find the period, amplitude, vertical displacement, and phase displacement of the graph of each equation. A. y 3cos(2 x ) 1 B. y 0.5sin(.5x .5 ) 3 C. y 3sin 2 x 60. State the type of symmetry: A. y x2 1 B. y x3 x 61. State the domain and range for: A. y sin x C. y x B. y cos x D. y sec x 62. State the domain and range for: A. y x C. y 3sec 2 x 1 6 B. y 3sin 2 x 1 4 1 3 D. y cot x 3 4 63. Find: csc 90 cot 0 sin180 cos 270 . 64. Graph y 3 cos 1 x 1. 2 3 65. Refer to the triangle to find exact value for the following: A. sin T = B. cos T = C. tan R = D. sec T = E. csc R = F. csc T = R 4 x 9 S0 6 G. cot R 66. Given: cos A 12 , csc B 13 A. cos A B B. sin 1 5 . Neither A nor B is in Quadrant III. Find: 3 B 2 C. tan 2A 67. Describe the transformations of each equation. A. B. C. D. f ( x) 2 cos x f ( x ) cos 21 x f ( x) cos( x 2 ) f ( x) cos x 2 68. Write an equation for the graph of the cosine function with the following characteristics: period of , phase shift left 2 , vertical stretch of 3 69. If f ( x) 2 x 4 and g( x) 3x 2 1 , find: A. f g x B. bg g f bg x C. f 2a D. g 5a b g 70. If g( x) x 2 1 and f ( x) x 2 , find f ( g( x)). T 71. Which graph is one-to-one? A. B. C. D. 72. Describe the graph as odd, even, both, or neither. A) ____________ B) ___________ C) ____________ 73. Name the asymptotes for y 2 sec x 3 . 74. Evaluate: csc14 to 4 decimal places. . . (in radians) 75. Find an acute angle such that tan 12617 Given: sin A 8 17 76. sin A B 1 77. cos B 2 , 2 A , tan B 7 24 , 3 2 B 2 . Find: 78. tan 2A Solve, 0 x 2 . 1 2 79. cos 2x sin x 1 80. sin 3 x 81. 2sin 2 x cos x 1 0 82. tan 2 x sec x 1 83. sin 2x 2cos x ________________________________________________________________________________ Solve for all values of x (general solution). 84. sin 2x sin x 85. cos 2x cos x sin 2x sin x 0 ________________________________________________________________________________ 86. A paddlewheel on a steamboat makes one complete revolution every 18 seconds. The diameter of the wheel is 10 feet. A point is at the bottom of the wheel, 2 feet below the surface of the water, at t = 0. Write an equation to represent the point's distance from the surface of the water. 87. A gear makes 6.2 rotations about its axis. what is the angular displacement in radians of a point on the gear? 88. What is the angular velocity in radians per minute of a notch on a wheel that makes 24 rotations per second about its axis? 89. The minute hand of a watch is 1.3 cm long. What is the linear velocity of the tip of the hand? Be sure to study: the graphs of the parent functions and trig functions with their domains and ranges the domain and range for all inverse trig functions the unit circle all of the trig identities