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AdvPreCal Spring Final Exam Review 1. Given sin x = 4/7 and cos x < 0, find cot x. 2. Simplify: csc π₯ πππ 2 π₯ 1+csc π₯ tan π₯ 3. Simplify: csc π₯ + sin π₯ tan π₯ 4. Simplify: tan π₯ 1βsec π₯ For 5-7, find all solutions in the interval ο0, 2ο° ο© : 6. sec2x β 3 tan x = 5 5. csc x + 2 = 0 8. Evaluate: sin 255β° 9. Evaluate: π‘ππ 7π 12 (Use the fact that 105ο° =210β°+45β°) (Use the fact that 10. Find the exact value: 7. 2 sin x cos x + cos x = 0 7π 12 tan 325°βπ‘ππ25° 1+π‘ππ325°π‘ππ25° = 3π 4 β π 6 . 11. Simplify: cos146β°cos11β°+sin146β°sin11β° 12. Given cot u = 2/5 , 0 < u < πΏ/2, and cos v = -3/5, πΏ < v < 3πΏ/2, find tan(u + v). 13. Given: sin x = -1/8 and tan x < 0, find sin 2x. 14. Given triangle with B = 87β°, C = 24β°, and a = 113, find b. 15. Given triangle with a = 18, b = 23, and B = 97β°, find C. 16. Given triangle with a = 72, b = 51, and A = 27β°, find the area of the triangle. 17. A hot air balloon is tethered to the ground with two 76 foot ropes. The angle of elevation each makes with the ground is 24β°. How far apart are the two ropes? 18. Given a triangle with a= 7, b=5, and c = 6, find A. 19. Given triangle with A = 38β°, b = 22, and c = 98, find a. 20. Use Heronβs formula to find the area of a triangle with a = 42, b = 51, and c = 57. 21. A boat leaves port and sails 16 miles at a bearing of S 20β° E. Another boat leaves the same port and sails 12 miles at a bearing of S 60β° W. How far apart are the two boats? 22. A vector v has initial point (3, 7) and terminal point (3, -2). Find its component form. 23. Determine the magnitude of the vector v: v = 7i β 2j. 24. Find the direction of v: v = β©3, β7βͺ 25. Given v of magnitude 50 and direction 315β°, and w of magnitude 20 and direction 206ο° , find v + w. 26. Given w = 2i β 3j and v ο½ ο2i ο« 5j , find w v . 27. Rewrite in trigonometric form: -2 + 3i 28. Rewrite in standard form: 5(cos 120β° + isin120β°) 29. Multiply: [3(cos85β° + isin85β°)][12(cos10β°+isin10β°)] 30. Evaluate: (3 +3i)8. 31. Find the number of distinguishable ways the letters OKEECHOBEE can be arranged. 32. Determine the number of seven digit telephone numbers that can be made under the condition that each of the first three digits cannot be zero. 33. An auto license plate is made using three letters followed by two digits. How many license plates are possible? 34. How many ways can 4 girls be selected from a group of 30 girls? 35. A box holds 11 white, 4 red, and 8 black marbles. If 2 marbles are picked at random, without replacement, what is the probability that they will both be black? 36. Find a formula for the nth term of the sequence. 3 4 5 6 {1 , 2 , 6 , 24 , β¦ } 37. Find the sum: 38. Find the 54th term of the sequence 10, 6, 2, -2, β¦ 39. Find the 24th term of the geometric sequence with a1 = 3 and r = 1.2. 40. Find the sum of the first 30 terms in the geometric sequence: 4, 26/5, 169/25, β¦ 41. Evaluate: 42. Evaluate: limπββ limπββ (1 + 2πβ8 π3 +11 2π4 β5π 3π4 +7πβ3 43. Evaluate: limπββ3 π₯ 3 +27 π₯+3 ) 45. Evaluate using the first 4 terms of the power series: cos 5π π₯ 2 +3π₯β88 (π₯+2)(π₯β3)2 48. Using the function π(π₯) = find the y-intercept(s). π₯ 2 +3π₯β88 find the x-intercept(s). (π₯+2)(π₯β3)2 49. Find the vertical asymptote(s) for f(x) = 50. Find the horizontal asymptote for f(x) = 51. Find the slant asymptote for f(x) = 9 π6 46. Evaluate using Eulerβs formula: 47. Using the function f(x) = π π₯β5 . π₯ 2 β6π₯+5 3π₯ 2 +8π₯β5 4π₯ 2 +π₯β6 . π₯ 2 +5π₯+4 π₯β3 52. Identify the removable discontinuity for π(π₯) = 2π₯ 2 +5π₯+3 π₯ 2 +π₯ 44. Evaluate: 53. Find the partial fraction decomposition of π₯β16 . π₯ 2 β2π₯β8 54. Find three other ways to represent the polar coordinate (3, 2πΏ/3) 55. Convert from rectangular to polar coordinates: (ββ3, β1) 56. Change from polar to a rectangular equation: rsinπΉ = -4. 57. Change from rectangular to a polar equation: x2 + y2 = 25 58. Graph each equation: a. r = 5 β 3sin πΉ b. π 2 = 16πππ 2π c. r = 2sin3πΉ d. r = 5 + 4sinπΉ 59. Find an equation for the graph at the right. a. r ο½ 3sin 4ο± b. r ο½ 3cos 2ο± c. r ο½ 3 ο 4cosο± d. r = 5cos2πΉ 60. Eliminate the parameters given x = 3t - 1 and y = t/4. 61. Write y = x2 β 4 in parametric form with the parameter t = x β 1. 62. Eliminate the parameters x = 4cos t and y = 5sin t. 63. Graph this parametric equation. T [-Ο, Ο] Tstep 0.1 x1 = 10cos(t) + 4cos(9t) y1 = 10sin(t) + 4sin(9t) X [-30, 30] Xscl 5 Y [--20, 20] Yscl 5