MATH 130 FINAL REVIEW

Transcription

1 MATH 130 FINAL REVIEW Problems 1 5 refer to triangle ABC, with C=90º. Solve for the missing information. 1. A = 40, c = 36m. B = 53 30', b = 75mm 3. a = 91 ft, b = 85 ft 4. B = 1, c = 4. ft 5. A = 66 54', b=8.8cm For problems 6 10, find the remaining trig functions of θ if: sinθ = and θ terminates in QII cosθ = and θ terminates in QIV. 5 1 tanθ = and terminates in QI cscθ = and cosθ < 0 5 cosθ = and terminates in QI For problems 11 and 1, simplify after making the given substitution x + 1, x = tanθ 5, = 5sin x x θ

2 For 13-15, perform the operation and simplify. Write answer in terms of sin θ and cosθ cosθ sinθ + sinθ cosθ 1 cosθ cosθ 15. ( cosθ For problems 16 and 17, write in terms of sin θ and cosθ and simplify. 16. secθ cotθ 17. cscθ cotθ cosθ For problems 18 and 19, give and angle between 0º and 360º coterminal with the given angle º º For problems 0 4, evaluate without using a calculator. Write your answers in radians arcsin 1 1. tan ( cos cos csc tan cos cos 3 1 5

3 For problems 5 and 6, θ is a central angle in a circle of radius r. Find the arc length s cut off by θ. 5. θ = 3, r = in 6. θ =, 1 3 r = cm 7. The minute hand of a clock is 1. cm long. How far does the tip of the minute hand travel in 40 minutes? For problems 8 and 9, find the area of a sector formed by a central angle θ in a circle of radius r. 8. θ =, r = 3 m 5 9. θ = 15, r = 10m 30. If the sector formed by central angle of 30º has an area of circle. 3 cm, find the radius of the For problems 31-35, identify the amplitude, period, phase shift, and vertical shift, and sketch the graph. 31. y = cos x 3. y = 3 + 3sin x y = tan x y = cos 3 x 35. y = csc x 4

4 3 3 For problems 36-41, let sin A = with A < and 5 Evaluate each of the following. 1 sin B = with B sin ( A + B 37. cos ( A B 38. cos B 39. sin B A 40. sin 41. cos A For problems 4 45, find exact values. 4. sin cos tan cos67.5 For problems 46 50, find all solutions in the interval 0 θ < 360. If necessary, round to the nearest tenth of a degree. 46. sinθ 1 = cosθ secθ = sinθ + cosθ = sin ( 3 45 θ = 3

5 50. tan θ = 1 For problems 51 54, find all solutions. Write your answer in radians using exact values. 51. cos x 3cos x = 5. 3 sin x cos x = sin x cos x + cos x sin x = sin 4x = 1 For problems 55 60, refer to triangle ABC, which is not necessarily a right triangle. 55. If B = 118, C = 37, and c =.9 in., find b. 56. If C = 60, a = 10 cm, and b = 1 cm, find c. 57. Find two triangles for which A = 51, a = 6.5 ft, and b = 7.9 ft. 58. Find two triangles for which A = 6, a = 4.8 ft, and b = 9.4 ft 59. Find all missing parts if a = 6.4 m, b =.8 m, and C = Find all missing parts if b = 3.7 m, c = 6. m, and A = 35 For problems 61 64, refer to triangle ABC. In each case, find the area of the triangle. Round the answer to the nearest hundredth. 61. a = 10 cm, b = 1 cm, C = B = 57, C = 31, a = 7.3m 63. B = 14 0', C = 75 40, b =.7 ft. 64. a = 8.3 ft, b = 6.3 ft, c = 3.45 ft

6 65. Combine: ( 7 + 3i + ( 4 i ( 3 + i 66. Simplify: 17 i 67. Multiply: ( 3 + 5i 68. Divide: i 6 5i For problems 69 and 70, write the answer in standard form: 69. 8( cos330 + i sin cis 135 For problems 71 and 7, write the answer in trigonometric form i i For problems 73 76, multiply or divide as indicated. Leave answers in trigonometric form ( cos 5 + i sin 5 3( cos 40 + i sin ( + i ( + i 10 cos50 sin 50 cos 0 sin ( cos10 + i sin [ 3cis 0 ] 4

7 For problems 77 80, use DeMoivre s Theorem. Write the answer in standard form ( cos15 + i sin ( 3cis ( 1+ i ( 1+ i Find two square roots of z 49( cos50 isin 50 form. = +. Leave your answers in trigonometric 8. Find 4 fourth roots of z = + i 3. Leave your answers in trigonometric form. 83. If the angle of elevation of the sun is 75º30, how tall is a post that casts a shadow 1.5 feet long? 84. A boat leaves the harbor entrance and travels 5 miles in the direction N 4º E. The captain then turns the boat 90º and travels another 18 miles in the direction S 48º E. At that time, how far is the boat from the harbor entrance, and what is the bearing of the boat from the harbor entrance? For problems 85 and 86, find the magnitude and direction of the vector , W = i + j For problems 87 and 88, find the components of the vector. 87. V = 13.8, θ = V = 64, θ = 0

8 MATH 130 FINAL REVIEW ANSWER KEY 1. B = 50, a = 3 m, b = 8m o '. A = 36 30, a = 536mm, c = 90mm 3. A = 47, B = 43, c = 15 ft 4. A = 69, a = 3.9 ft, b = 1.5 ft 5. B = 3 06', a = cm, c = 7.08cm cos θ =, tan θ =, csc θ =, sec θ =, cotθ = sin θ =, tan θ =, csc θ =, sec θ =, cot = sin θ =, cos θ =, csc θ =, sec θ =, cotθ = sin θ =, cosθ =, tan θ =, sec θ =, cotθ = sin θ =, tanθ = 1, cscθ =, secθ =, cotθ = secθ 1. 5 cosθ sinθ cosθ 14. sin θ cosθ cos θ 4cosθ sin θ

9 17. sinθ 18. 5º º in cm cm or 1.6 cm m 5 m cm 31. Amplitude = 1, Period =, Phase shift =

10 3. Amplitude = 3, Period = 6, Phase Shift = 1, Vertical shift = Period =, Phase shift = Amplitude = 1, Period =, Phase shift = 3 6

11 35. Amplitude = 1, Period =, Phase shift = º, 150º

12 47. 0º, 180º 48. 0º, 90º º, 115º, 15º, 35º, 335º, 355º 50..5º, 11.5º, 0.5º,9.5º k, + k, + k k, + k k k in cm 57. B = 71, C = 58, c = 7.1 ft, B ' = 109, C ' = 0, c ' =.9 ft 58. B = 59, C = 95, c = 11 ft, B ' = 11, C ' = 33, c ' = 6.0 ft 59. A = 44, B = 17, c = 8.1m o o 60. B = 34, C = 111, a = 3. 8m cm m ft ft I

13 i 87. V = 1.6, V = 5.66 x y i 88. Vx = 64, Vy = i i 71. ( cos 45 + i sin ( cos150 + isin ( cos65 + i sin ( cos30 + i sin ( cos50 + isin cis i i i cis5, 7cis05 8. cis15, cis105, cis195, cis ft miles, bearing N 77.8º E W = 5