1. Determine the remaining sides and angles of the triangle ABC. Show all work and / or support your answer.

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1 Trigonometry Final Exam Review: Chapters 7, 8 Chapter 7: Applications of Trigonometry and Vectors 1. Determine the remaining sides and angles of the triangle ABC. 2. Determine the remaining sides and angles of the triangle ABC. C = 71.83, B = 42.57, a = A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km distant is N 38.8 E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2 E. How far did the ship travel between the two observations of the lighthouse? 4. Mark notices that the bearing of a tree on the opposite bank of a river flowing north is Lisa is on the same bank as Mark, but m away. She notices that the bearing of the tree is The two banks are parallel. What is the distance across the river? 5. Find the area of the triangle ABC. A = 30.50, b = cm, C = A real estate agent wants to find the area of a triangular lot. A surveyor takes measurements and finds that two sides are 52.1 m and 21.3 m, and the angle between them is What is the area of the triangular lot? Show all work and / or support 7. Determine the number of triangles ABC possible with the given parts. a) b = 60, a = 82, B = 100 b) a = 35, b = 30, A = 40 c) B = 54, c = 28, b = Find the unknown angles in triangle ABC, if the triangle exists. a) C = 4120', b = 25.9 m, c = 38.4 m b) B = 48.2, a = 890 cm, b = 697 cm c) B = 74.3, a = 859 m, b = 783 m 9. Solve the triangle ABC, if possible. C = 52.3, a = 32.5 yd, c = 59.8 yd 10. Solve the triangle ABC, if possible. A = 3840', a = 9.72 km, b = 11.8 km 11. Solve each triangle and then find the area of the triangle. Approximate values to the nearest tenth. Show all work 1

2 and / or support a) b) 12. Solve each triangle and then find the area of the triangle. a) C = 28.3, b = 5.71 in., a = 4.21 in. b) a = 189 yd, b = 214 yd, c = 325 yd 13. The sides of a parallelogram are 4.0 cm and 6.0 cm. One angle is 58 while another is 122. Find the lengths of the diagonals of the parallelogram. 14. A weight is supported by cables attached to both ends of a balance beam, as shown in the figure. What angles are formed between the beam and the cables? 15. Using the vectors in the figure below, draw a sketch to represent: a) c + d b) d c c) 3c 16. Use the vectors in the adjacent figure to find a) a + b, b) a b, c) a. 17. For vectors u and w with angle θ between them, sketch the resultant, u + w. u 8, w 12, θ = Find the magnitude and direction angle for vector v = 15, Vector v has magnitude v and direction angle α Find the horizontal (a) and vertical (b) components of v. 2

3 20. Write vector v in component form, a, b. 21. Two forces act at a point in the plane. One is 19 newtons, the other is 32 newtons. The angle between the forces is 118. Find the magnitude of the resultant force. 22. Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the adjacent figure. Round your answer to the nearest tenth. 23. Given u = 2, 5 and v = 4, 3, find 2u + 4v. Write your answer in the form ai + bj. 24. Find the dot product of 2, - 3 and 6, Find the angle between the vectors -5i + 12j and 3i + 2j. 26. Given u = 2, 1, v = 3, 4, and w = 5, 12, evaluate u v w. 27. Determine whether the vectors 1, 2 and 6, 3 are orthogonal. 28. Two tugboats are pulling a disabled speedboat into port with forces 1240 lb and 1480 lb. The angle between these forces is Find the magnitude and direction of the equilibrant. 29. Two forces of 128 lb and 253 lb act at a point. The equilibrant force is 320 lb. Find the angle between the forces. 30. A force of 176 lb makes an angle of 7850' with a second force. The resultant of the two forces makes an angle of 4110' with the first force. Find the magnitudes of the second force and the resultant. Show all work and / or support 31. A 130-lb force keeps a 600-lb object from sliding down an inclined ramp. Find the angle of the ramp. 32. Find the magnitude of force necessary to keep a 3500-pound car from sliding down a ramp of Two people are carrying a box. One person exerts a force of 150 lb at an angle of 62.4 with the horizontal. The other person exerts a force of 114 lb at an angle of Find the weight of the box. Show all work and / or support 3

4 34. A ship leaves port on a bearing of 34.0 and travels 10.4 miles. The ship then turns due east and travels 4.6 miles. How far is the ship from port, and what is its bearing from port? 35. A pilot is flying at 168 mph. She wants her flight path to be on a bearing of 5740'. A wind is blowing from the south at 27.1 mph. Find the bearing the pilot should fly, and find the plane s groundspeed. Show all work and / or support 36. A pilot wants to fly on a bearing of By flying due east, he finds a 42-mph wind, blowing from the south, puts him on course. Find the groundspeed and airspeed. Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations 37. Write 288 as the product of a real number and i. 38. Solve 3x 2 2 4x and express all nonreal complex solutions in terms of i. Show all work and / or support your answer. 39. Perform the indicated operations and then simplify. a) b) Perform the indicated operation and then simplify. Write your answers in standard form. Show all work and / or support a) 4 i 2 3i 4 5i b) 2 3i 4 2i c) 2 3i 1 5i 41. Simplify each power of i. a) 25 i b) 13 i 42. Find the sum of 5 6i and 2 3i by graphing each number in the complex plane and finding their resultant. 43. Write the complex number 3[cos i sin 150] in rectangular form. 44. Write the complex number 5 5i in trigonometric form r(cos θ + i sin θ), with θ in the interval [0, 360). 45. Find each product and write it in rectangular form. On part b, round to four decimal places. Show all work and / or support a) 8 cos 300 i sin cos120 i sin 120 b) (12 cis 18.5) (3 cis 12.5) 4 cos120 i sin Find the quotient 2 cos150 i sin 150 & write it in rectangular form. 4

5 2i 47. Convert the numerator and denominator of to trigonometric form. Then find the quotient and write it in 1 i 3 rectangular form. 48. Find each power. Write each answer in rectangular form. a) 3 cis100 3 b) 2 2i Find all cube roots of each complex number. Leave answers in trigonometric form. Show all work and / or support a) 27 cis 300 b) 3 i 50. Find all complex solutions of each equation. Leave answers in trigonometric form. Show all work and / or support a) x 4 i 0 b) x For each pair of polar points, plot the point; give two other pairs of polar coordinates (including one with a negative r); and give the rectangular coordinates for the point. 5 b) 3, 3 a) 2, Given the rectangular coordinates 3 1,, give two pairs of polar coordinates for the point, including one with a 2 2 negative r. Select θ so it is in the interval [0, 360). 53. For each rectangular equation, give its equivalent polar equation and sketch its graph. Show all work and / or support 2 2 a) 3x 2y = 6 b) x y 9 5

6 54. Match each equation with its corresponding graph. a. r = a cos θ A. b. r = a sin θ B. a b 1 c. r = a ± b sin θ with C. a b 1 d. r = a ± b cos θ with D. e. r = a cos 4θ E. f. r = a sin 5θ F. c acos θ b sin θ g. r = G. h. r 2 = 4 sin 2θ H. 55. Identify the type of polar equation and draw a complete graph. Include a t-chart with at least 5 key points with your graph. 6

7 a) r = cos θ b) r = 4 cos 2θ 56. For each polar equation, give its equivalent rectangular equation and sketch its graph. Include a t-chart with at least 5 key points with your graph. 3 a) r = 2 sin θ b) r 1 sinθ Answers Chapter 7: Applications of Trigonometry and Vectors 1. A = 37.2, a = 178 m, c = 244 m 2. A = 65.60, b = cm, c = cm km m cm m 2 7. a) 0; b) 1; c) 2 8. a) B 2630', A 11210'; b) A , C , A , C ; c) no such triangle 9. A 25.5, B 102.2, b 73.9 yd 10. B ', C ', c km, B ', C ', c km 7

8 11. a) A 22.3, B 108.2, C 49.5; area = units b) A 56.7, C 68.3, b 88.2; area = units 12. a) c 2.83 in., A 44.9, B 106.8; area = units b) C 10720', B 3900', A 3340'; area = units cm, 8.8 cm and a) b) c) 16. a) 0, 10 b) 8, 2 c) 4, v 17 ; θ = a 13.7, b v = 4, newtons lb i + 2j orthogonal lb at an angle of with the 1480-lb force magnitude of second force: 190 lb; magnitude of the resultant: 283 lb lbs lb mi; bearing: 6530'; groundspeed: 181 mph 36. groundspeed: 161 mph; airspeed: 156 mph 8

9 Chapter 8: Complex Numbers, Polar Equations, and Parametric Equations i x i a) i a) 10 i b) 3 b) 2 16i c) 41. a) i b) i i ; i cos225 i sin a) 20 20i 3 b) i i i a) i b) i a) 3 cis 100, 3 cis 220, 3 cis 340 b) 3 2 cos 110 i sin 110, 3 2 cos 230 i sin cos 350 i sin 350, 50. a) cos i sin 22.5, cos i sin 112.5, cos i sin 202.5, cos i sin b) 2 cis 45, 2 cis 135, 2 cis 225, 2 cis a) 2, 495, (2, 315), 2, 2 π 4π b) 3,, 3,, 3 3 3, (1, 210), (-1, 30) 53. a) 6 r b) r = 3 or r = 3 3cosθ 2sinθ 54. a. F b. A c. D d. G e. H f. B g. C h. E 9

10 55a. 55b. 56a. 56b. 10