Math Trigonometry Exam with Answer Key

Pre-Calculus Final Exam Review – 2nd semester TRIGONOMETRY Solve for 0    2 without using a calculator: 1. sin    1 2 2. csc  3. tan   1 2 4. cos  2 1)________________ 2)________________ 3)________________ 4)________________ Solve for  in degrees giving all solutions. 5. sin  1 6. cos   3 2 7. tan   undefined 5)________________ 6)________________ 7)________________ Give the exact value of each expression. 8. Tan 1 3 9. 1  cot Sin 1  2  10.   3  sec Cos 1      5   8)________________ 9)________________ 10)_______________ 11. Given y  1 1 sin x , find 3 2 a. amplitude 11a.)_____________ b. period 11b.)_____________ c. graph at least two periods of the function   12. Given: y  4  2 cos 3 x  a. b. c. d. e.   , find 2 12a)______________ Amplitude Period Phase shift Vertical shift graph at least two periods of the function 12b)______________ 12c)______________ 12d)______________ 1  sin 1  csc sin 13. Simplify: 13)_______________ 14. Prove: 15. Solve: 2 cos2   1  sin for  if 0    2 . 16. Solve for x: 17. Solve a right triangle ABC, if a = 6 and A  300 . (Give exact lengths.) sec  csc  csc 1  tan  15)_______________ 2 cos x  1  0 , 0  x  2 . 16)_______________ 17) B  __________ b = _________ c = __________ 18. A boy flying a kite is standing 20 ft from a point directly under the kite. If the string to the kite is 40 ft long, find the angle of elevation of the kite. 18)_______________ 19. An airplane is at an elevation of 30,000 ft and approaches the airport with an angle of descent of 5°. What is the distance between the airport and the point on the ground directly below the airplane? 19)_______________ 2 20. Given ∆ABC with a = 30, b = 20, c = 40, find the largest angle. 20)_______________ 21. The captain of a clipper ship spots two other ships on the ocean. One ship is 5 miles away while the other is 5.2 miles away. The angle between the two sightings is 20º. How far apart are the two observed ships? (to three decimals) 21)_______________ 22. In ∆RST, R  1370 ,t = 15, and s = 12. Find r to the nearest integer. 22)_______________ 23. If ∆ABC has b = 30,  C = 40 0 and  A = 60 0 , find a to the nearest tenth. 23)_______________ 24. Two angles of a triangle measure 29º and 51º. The longest side is 55cm. Find the length of the shortest side to the nearest tenth. 24) ______________ 25. Solve for  ABC if a = 15, c = 18, and A = 32º. 25) C= __________ B= __________ b= __________ OR (if two triangles) C= __________ B= __________ b= __________ 26. Find the area of ∆ABC if b = 32, c = 27, and  A = 108º. 26)___ ____ 27. The area of ∆PQR is 15. If p = 5 and q= 10, find all possible measures of  R. 27)___ ____ 3 28. Find the area of a regular pentagon inscribed in a circle with radius 15. 28) ____ 29. The sides of an isosceles triangle have lengths 7, 10 and 10. What are the measures of its angles? 29.)______________ 30. At a distance of 200 meters, the angle of elevation to the top of a building is 70. Approximately how tall is the building? 30)_______________ 31. Two ships leave a port on courses that differ by 70 and each travels at 25 knots. In terms of nautical miles, how far apart are the ships after 1 hour? 31)_______________ 32. After leaving an airport, a plane flies for 1.75 hours at a speed of 200 k/h on a course of 100. The plane then flies for 2 hours at a speed of 250 k/h on a course of 40. At this time, how far from the airport is the plane? 33. 32.)______________ Find the exact value of the following: a. cos 75 b. sin 105 33a.)_____________ 33b.)_____________ 34. Simplify the following: a. cos75 cos15  sin 75sin15 sin(30  x)  sin(30  x) b. 34a.)_____________ 34b.)_____________ 35. Suppose angle A is acute and cos A  a. sin A b. cos 2 A 5 . Find: 13 sin 2 A c. 35a.)_____________ 35b.______________ 35c.)_____________ 36. Simplify the following: a. 1 cos 2 x sin 2 x b. 1  cot x cos 2 x  1 2 c. tan x sec x  1 36a.)_____________ 36b.)_____________ 36c.)_____________ 4 37. Evaluate the given expression: 38. Prove: 1  tan 2 x 1  cos 2 x   2 39. Solve the following for 0  x  2 .  a. 5 12 37.)______________ 2 sin2 x  3 cos x  3 a.)_______________ 1  2 sin2  cos 2x  sin x  2 b. b.)_______________ c. sin x tan x  2 sin x c.)_______________ POLAR COORDINATES AND COMPLEX NUMBERS 40. Convert the following into rectangular form. a. 6, 90 b.    2,   6 40a.)_____________ 40b.)_____________ 41. Convert the following into polar form: a. (3,3) b. (-1,  3 ) c. (0, -2) 41a.)_____________ 41b.)_____________ 41c.)_____________ 42. If z1  3  i and z 2  4  4i , find z1, z2 , and z1z2 in polar form. 42.) z1  __________ z2  _________ z1z2 = ___________ 43. If z = (4, 30°), find the following in a  bi form: a. z3 b. z5 c. 43a) z -2 43b) 43c) 44. Find the cube roots of 3i . 44.)____________________ _______________________ _______________________ 5 SEQUENCES AND SERIES 45. State whether each sequence is arithmetic, geometric or neither and find a formula for a n . a. 17, 12, 7, 2, … b. 3, 8, 15, 24, 35,… c. -81, 27, -9, 3,… 45a.)___________________ _______________________ 45b.)___________________ _______________________ 45c.)___________________ _______________________ 46 a. Find the 17th term of 3, 6, 9, … 46a.)___________________ b. Find the sum of the first 17 terms of 3, 6, 9, … 46b.)___________________ 4 . Find a10 . 27 47. In a geometric sequence, a3  4 and a6  48. In an arithmetic sequence, a3  23 and a6  50 . Find a24 . 49. An auditorium has 30 rows of seats. There are 20 seats in the first row, 24 seats in the second row, 28 seats in the third row, and so on. Determine the seating capacity in the auditorium. 49.)____________________ 50. Given the series 1  51. 47.)____________________ 48.)____________________ 1 1 1    3 9 27 a) Does it converge or diverge? 50a) b) Find the sum, if possible. 50b) 1 1 1 ,  , 3 9 27 Find the sum of the first 8 terms of 1,  , 6 51) 52. Find the interval of convergence and the sum in terms of x of: 1 2x 4x2  ... 3 9 52.)____________________ _______________________ 53. 54. 55. Express the following series in sigma notation: 8 + 5 + 2 – 1 – 4 – 7 – 10 – 13 53.)____________________ t1  1 t k 1  t k  2k  1 a) Write the first 6 terms 54a) b) Write an explicit formula 54b) Evaluate the following: 7 a.  ( 4n  7) 55a.)___________________ n 3 20 b.  2k (k  1) 55b.)___________________ k 1 56. Express in sigma notation: 5 + 9 + 13 + …+ 101 57. For what values of x do the following converge? 56.) a) 1 + (x-3) + (x-3)2 + … 57a) b) 1 + 3x + 9x2 + … 57b) 7 58. 59. 60. Given the series 1  a) Express the series in sigma notation. 58a) b) Find the sum. 58b) Write a recursive definition for the following sequence: 6, 10, 14, 18, 22, … 59) For what value of x does the following sequence converge 60) to 61. 1 1 1    3 9 27 3 ? 5 1 + 2x + 4x2 + … Prove by Mathematical Induction. n  4i  1  n 2n  1 i 1 8 LIMITS, DERIVATIVES AND APPLICATIONS OF DERIVATIVES (MAX/MIN PROBLEMS) Find the following limits, if they exist. If they do not exist, write “does not exist.” 62. 3n lim n  2 62) n 63. x 2 64. x2 2 4 lim x lim x 7 3 x  2 2x 65. 2n 2  n lim n 1 n  66. lim  1  n n  67. 63) 64) 65)  3n  2   n 1  66) x3 lim x  3 67) x 3 68. lim x 1 69. lim 1  x 0 70. 5 x 2  3x  1 x2  5 68) x 69) 1 x lim sin n lim 1 1 x 1 x 70) n  71. x 0 71) 9 72. lim 3  x 2 73. lim 3n 2  n 5n 2  1 73) lim n 2  2n n2  4 74) lim 5n 2  3n  1 n2  5 75) n  74. n 2 75. n  76. lim n  77. 72) 1 x x 1 lim x 2 n3 76)____________________ n2  9 x3  8 x2 77)____________________ Find the derivative of the following (using the special rules/techniques). 78. f ( x)  4 x 5  2 x 3  9 x 1  5 79. f ( x)  80. f ( x)  81. f ( x)  78.)____________________ 1 79.)____________________ 3x  12 6 x2 80.)____________________ x 1 81.)____________________ x 10    82. f ( x)  x 3  2 x 3x 2 83. f ( x)  84. f ( x)  3x2  4 x  1 85. 3x x 2  2 x  15 f ( x)  x2  9 85.)____________________ 86. f ( x)  3 3 x  86.)____________________ 82.)____________________ x2  4 x2  4x  4  83.)____________________   2 84.)____________________  7 Use the difference quotient, lim h0 f ( x  h)  f ( x) , to find the derivative. h 87. f ( x)  5  6x 87.)____________________ 88. f ( x)  x 2  3 x  5 88.)____________________ 89. f ( x)  3x 2  4 x  6 89.)____________________ 11 Find the slope of the graph at the given point. Use the result to find an equation of the tangent line to the graph at the point. 90. f ( x )  x 2  1 ; (2, 3) 90.)____________________ _______________________ 91. f ( x)  x 3  x ; (2, 6) 91.)____________________ _______________________ 92. f ( x)  x  1 ; (3, 2) 92.)____________________ _______________________ 93. f ( x)  2 x  4 ; (2, 6) x 93.)____________________ _______________________ 94. Find the equation of the tangent to the given curve: y  x 3  5 x 2  4 x  2 ; when x  2 94.)____________________ 95. If f ' ( x)  3 x 2  4 x  3 , find f (x) . 95.)____________________ ****************************************************************************************************************************** 96. Use the first and second derivative to identify the local max and min, inflection point/s and determine the intervals where the curve is concave up and concave down. Then graph the function. (DO NOT USE A GRAPHING CALCULATOR). f ( x)  x 3  3x 2  9 x  27 Local Max_______________ Local Min_______________ Pt. of Inflection_________________ Interval/s: Concave Up___________________ Concave Down_________________ 12 97. A ball is thrown upward from the top of an 80 ft building so that its height in feet above the ground after t seconds is h(t )  80  64t  16t 2 . a. What is the instantaneous velocity at t  1 second? 97a.)___________________ b. When is the velocity = 0? 97b.)___________________ c. What is the ball’s maximum height above the ground? 97c.)___________________ d. When does the ball hit the ground? 97d.)___________________ e. For what values of t is the ball falling? 97e.)___________________ Use derivatives to solve. 98. The number 120 is divided into two parts such that the product of one number times the square of the other is a maximum. Determine the two numbers. 98.)____________________ 99. 800 yards of fencing is used to enclose a rectangular field with a fence down the middle parallel to one of the sides. What is the maximum area which can be enclosed? 99.)____________________ 100. A cardboard poster is to have 50 square inches of printed material surrounded by a 2” border at the top, 2” at the bottom and 1” on each side. Find the minimum dimensions of the poster which has a minimum area. 100.)_____________________ 101. An open square-base box is to be manufactured from the least amount of material. If the box is to have a volume of 32 cubic meters, what dimensions will minimize the amount of material used? 101.)___________________ 13 Pre-Calculus Final Exam Review Packet Answer Key 1. 7 11 , 6 6 2. 5. 270  360n 6. 3 9.  3 4 , 4 150  360n 210  360n 10.  5 3 3. 3 7 , 4 4 4. 7. 90  180n 8. 11a. 1 3 11b. 4 12d. -4  undefined  3 , 60 12a. 2 12b. 2 3 12c. 13. cos2  14. answers will vary 15. 17. B  60; b  6 3; c  12 20. 104.5 21. 1.782 miles 24. 27.1 25. C  39; B  109; b  27.0 or C  141; B  7; b  3.5 26. 410.86 27. 36.9 or 143.1 28. 534.97 29. 30. 549.5 m 31. 28.7 32. 739.93 miles 33a. 6 2 4 34a. cos90  0 34b. cos x 35a. 12 13 36a. cot x 36b. 2 cot2 x 38. answers will vary 39a. 40a. 6i 40b. 41c. 3   2 ,  2   43a. 64i 33b. 35b.  119 169 35c. 120 169 36c. sin x 1  cos x 37.  39b. , 39c. 0 ,  , 1.11, 4.25 41a. 42. 43c. 2 4 , 3 3 3 2 4     41b.  2 ,  3 2 ,  3  4   11   z1  2 cis ; z 2  4 2 cis ; z1 z 2  8 2 cis 6 4 12 1 3  i 32 32 44. 3 2  5 3 6 6 , 2 16.  7 4 , 4 18. 60 19. 342901.6 ft. 22. 25 23. 26.4 2  cis10, 3 2  cis130, 3 2  cis 250 14 , 69.5, 69.5, 41 6 2 4  2 3i 43b.  512 3  512i 45a. arith a n  5n  22 45b. neither; an  n  2n 45c. 46b. 459 47. 50a. converge 50b. 52.  2 3 3 3  x ; 2 2 3  2x 1 geometric; a n  81    3 4 2187 3 4 n 1 46a. 51 49. 2340 54b. an  n 2  4n  1 57a. 2 x  4 58b. 3 2 59. 63. 48. 212 51. 1640 2187   3n  11 54a. 1, 4, 9, 16, 25, 36 5320 56. 8 53. i 1 25 55a. 65 55b. i 1 57b. 60. 1 1   x 3 3 1  3 64. 0 68.  3 4 3 6 72. 76. 1 79.  6 3x  1 3  n 1 a1  6 58a. 1    i 1  3  61. omit!! 62. 3 65.  66. does not exist (oscillates) 67.  69. 2 70. does not exist (oscillates) 71. does not exist 73. 3 5 74. 1 2 75. 5 77. 12 78.  20x 6  6x 4  9x 2 80.  82. 15x 4  18x 2 86. 73x  3 12 x3 81. x 1 2x x   3 x 2  6 x  15 a n  a n1  4 1  4 x  2 84. 12x  83x 2  4 x  1 85. 87. -6 88. 2x – 3 89. 6x + 4 90. 4; y  4x  5 91. 11; y  11x  16 92. 1 1 5 ; y  x 4 4 4 93. 1; y  x  4 94. y  4 x  6 32 97b. 2 sec. 97c. 144 ft 97d. 5 sec. 97e. 2 t  5 98. 80 and 40 99. 26,666 83. 95. 96. 4 2 f ( x)  x 3  2 x 2  3x  c Local Max=(-1,32) 97a. Local Min= (3,0) Point of inflection=(1,16) Concave Up: (1,∞) Concave Down: (-∞,1) 100. x  32 4 7 inches x 14 inches 101. 4x4x2 15 2 yds2 3