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Review Exercises

Determine whether the statement is true or false.

1. For any point $(x, y)$ $(x, y)$ on the unit circle, $〈 x, y 〉$ $〈 x, y 〉$ is a unit vector. [8.6]
2. The law of sines can be used to solve a triangle when all three sides are known. [8.1]
3. Two vectors are equivalent if they have the same magnitude and the lines that they are on have the same slope. [8.5]
4. Vectors $〈 8, - 2 〉$ $〈 8, - 2 〉$ and $〈 - 8, 2 〉$ $〈 - 8, 2 〉$ are equivalent. [8.6]
5. Any triangle, right or oblique, can be solved if at least one angle and any other two measures are known. [8.1]
6. When two angles and an included side of a triangle are known, the triangle cannot be solved using the law of cosines. [8.2]

Solve $Δ A B C$ $Δ A B C$ , if possible. [8.1], [8.2]

7. $a = 23.4 ft, b = 15.7 ft, c = 8.3 ft$ $a = 23.4 ft, b = 15.7 ft, c = 8.3 ft$
8. $B = 27 °, C = 35 °, b = 19 in$ $B = 27 °, C = 35 °, b = 19 in$ .
9. $A = 133 ° 28', C = 31 ° 42', b = 890 m$ $A = 133 ° 28', C = 31 ° 42', b = 890 m$
10. $B = 37 °, b = 4 yd, c = 8 yd$ $B = 37 °, b = 4 yd, c = 8 yd$
11. Find the area of $Δ A B C$ $Δ A B C$ if $b = 9.8$ $b = 9.8$ , $c = 7.3 m$ $c = 7.3 m$ , and $A = 67.3 °$ $A = 67.3 °$ . [8.1]
12. A parallelogram has sides of lengths 3.21 ft and 7.85 ft. One of its angles measures $147 °$ $147 °$ . Find the area of the parallelogram. [8.1]
13. Sandbox. A child-care center has a triangular-shaped sandbox. Two of the three sides measure 15 ft and 12.5 ft and form an included angle of $42 °$ $42 °$ . To determine the amount of sand that is needed to fill the box, the director must determine the area of the floor of the box. Find the area of the floor of the box to the nearest square foot. [8.1]
14. Flower Garden. A triangular flower garden has sides of lengths 11 m, 9 m, and 6 m. Find the angles of the garden to the nearest degree. [8.2]
15. In an isosceles triangle, the base angles each measure $52.3 °$ $52.3 °$ and the base is 513 ft long. Find the lengths of the other two sides to the nearest foot. [8.1]
16. Airplanes. Two airplanes leave an airport at the same time. The first flies $175 km/h$ $175 km/h$ in a direction of $305.6 °$ $305.6 °$ . The second flies $220 km/h$ $220 km/h$ in a direction of $195.5 °$ $195.5 °$ . After 2 hr, how far apart are the planes?[8.2]

Graph the complex number and find its absolute value. [8.3]

17. $2 - 5 i$ $2 - 5 i$
18. 4
19. 2i
20. $- 3 + i$ $- 3 + i$

Find trigonometric notation. [8.3]

21. $1 + i$ $1 + i$
22. $- 4 i$ $- 4 i$
23. $- 5 \sqrt{3} + 5 i$ $- 5 \sqrt{3} + 5 i$
24. $\frac{3}{4}$ $\frac{3}{4}$

Find standard notation, $a + b i$ $a + b i$ . [8.3]

25. $4 (\cos 60 ° + i \sin 60 °)$ $4 (\cos 60 ° + i \sin 60 °)$
26. $7 (\cos 0 ° + i \sin 0 °)$ $7 (\cos 0 ° + i \sin 0 °)$
27. $5 (\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})$ $5 (\cos \frac{2 π}{3} + i \sin \frac{2 π}{3})$
28. $2 [\cos (- \frac{π}{6}) + i \sin (- \frac{π}{6})]$ $2 [\cos (- \frac{π}{6}) + i \sin (- \frac{π}{6})]$

Convert to trigonometric notation and then multiply or divide, expressing the answer in standard notation. [8.3]

29. $(1 + i \sqrt{3}) (1 - i)$ $(1 + i \sqrt{3}) (1 - i)$
30. $\frac{2 - 2 i}{2 + 2 i}$ $\frac{2 - 2 i}{2 + 2 i}$
31. $\frac{2 + 2 \sqrt{3} i}{\sqrt{3} - i}$ $\frac{2 + 2 \sqrt{3} i}{\sqrt{3} - i}$
32. $i (3 - 3 \sqrt{3} i)$ $i (3 - 3 \sqrt{3} i)$

Raise the number to the given power and write trigonometric notation for the answer. [8.3]

33. ${[2 (\cos 60 ° + i \sin 60 °)]}^{3}$ ${[2 (\cos 60 ° + i \sin 60 °)]}^{3}$
34. ${(1 - i)}^{4}$ ${(1 - i)}^{4}$

Raise the number to the given power and write standard notation for the answer. [8.3]

35. ${(1 + i)}^{6}$ ${(1 + i)}^{6}$
36. ${(\frac{1}{2} + \frac{\sqrt{3}}{2} i)}^{10}$ ${(\frac{1}{2} + \frac{\sqrt{3}}{2} i)}^{10}$
37. Find the square roots of $- 1 + i$ $- 1 + i$ . [8.3]
38. Find the cube roots of $3 \sqrt{3} - 3 i$ $3 \sqrt{3} - 3 i$ . [8.3]
39. Find and graph the fourth roots of 81. [8.3]
40. Find and graph the fifth roots of 1. [8.3]

Find all the complex solutions of the equation. [8.3]

41. $x^{4} - i = 0$ $x^{4} - i = 0$
42. $x^{3} + 1 = 0$ $x^{3} + 1 = 0$
43. Find the polar coordinates of each of these points. Give three answers for each point. [8.4]

Find the polar coordinates of the point. Express the answer in degrees and then in radians. [8.4]

44. $(- 4 \sqrt{2}, 4 \sqrt{2})$ $(- 4 \sqrt{2}, 4 \sqrt{2})$
45. $(0, - 5)$ $(0, - 5)$

Convert from rectangular coordinates to polar coordinates. Express the answer in degrees and then in radians. [8.4]

46. $(- 2, 5)$ $(- 2, 5)$
47. $(- 4.2, \sqrt{7})$ $(- 4.2, \sqrt{7})$

Find the rectangular coordinates of the point. [8.4]

48. $(3, \frac{π}{4})$ $(3, \frac{π}{4})$
49. $(- 6, - 120 °)$ $(- 6, - 120 °)$

Convert from polar coordinates to rectangular coordinates. Round the coordinates to the nearest hundredth. [8.4]

50. $(2, - 15 °)$ $(2, - 15 °)$
51. $(- 2.3, \frac{π}{5})$ $(- 2.3, \frac{π}{5})$

Convert to a polar equation. [8.4]

52. $5 x - 2 y = 6$ $5 x - 2 y = 6$
53. $y = 3$ $y = 3$
54. $x^{2} + y^{2} = 9$ $x^{2} + y^{2} = 9$
55. $y^{2} - 4 x - 16 = 0$ $y^{2} - 4 x - 16 = 0$

Convert to a rectangular equation. [8.4]

56. $r = 6$ $r = 6$
57. $r + r \sin θ = 1$ $r + r \sin θ = 1$
58. $r = \frac{3}{1 - \cos θ}$ $r = \frac{3}{1 - \cos θ}$
59. $r - 2 \cos θ = 3 \sin θ$ $r - 2 \cos θ = 3 \sin θ$

In Exercises 60–63, match the equation with one of figures (a)–(d) that follow. [8.4]

60. $r = 2 \sin θ$ $r = 2 \sin θ$
61. $r^{2} = \cos 2 θ$ $r^{2} = \cos 2 θ$
62. $r = 1 + 3 \cos θ$ $r = 1 + 3 \cos θ$
63. $r \sin θ = 4$ $r \sin θ = 4$

Magnitudes of vectors u and v and the angle $θ$ $θ$ between the vectors are given. Find the magnitude of the sum, $u + v$ $u + v$ , to the nearest tenth and give the direction by specifying to the nearest degree the angle that it makes with the vector u. [8.5]

64. $| u | = 12$ $| u | = 12$ , $| v | = 15$ $| v | = 15$ , $θ = 120 °$ $θ = 120 °$
65. $| u | = 41$ $| u | = 41$ , $| v | = 60$ $| v | = 60$ , $θ = 25 °$ $θ = 25 °$

The vectors u, v, and w are drawn below. Copy them on a sheet of paper. Then sketch each of the vectors in Exercises 66 and 67. [8.5]

66. $u - v$ $u - v$
67. $u + \frac{1}{2} w$ $u + \frac{1}{2} w$
68. Forces of 230 N and 500 N act on an object. The angle between the forces is $52 °$ $52 °$ . Find the resultant, giving the angle that it makes with the smaller force. [8.5]
69. Wind. A wind has an easterly component of $15 km/h$ $15 km/h$ and a southerly component of $25 km/h$ $25 km/h$ . Find the magnitude and the direction of the wind. [8.5]
70. Ship. A ship sails $N 75 ° E$ $N 75 ° E$ for 90 nautical mi, and then $S 10 ° W$ $S 10 ° W$ for 100 nautical mi. How far is the

ship, then, from the starting point and in what direction? [8.5]

Find the component form of the vector given the initial and terminal points. [8.6]

71. $\overset{⇀}{A B}; A (2, - 8), B (- 2, - 5)$ $\overset{⇀}{A B}; A (2, - 8), B (- 2, - 5)$
72. $\overset{⇀}{T R}; R (0, 7), T (- 2, 13)$ $\overset{⇀}{T R}; R (0, 7), T (- 2, 13)$
73. Find the magnitude of vector u if $u = 〈 5, - 6 〉$ $u = 〈 5, - 6 〉$ . [8.6]

Do the calculations in Exercises 74–77 for the vectors

u = 〈 3, - 4 〉, v = 〈 - 3, 9 〉, and w = 〈 - 2, - 5 〉 .

$u = 〈 3, - 4 〉, v = 〈 - 3, 9 〉, and w = 〈 - 2, - 5 〉 .$

[8.6]

74. $4 u + w$ $4 u + w$
75. $2 w - 6 v$ $2 w - 6 v$
76. $| u | + | 2 w | $ $| u | + | 2 w | $
77. $u \cdot w$ $u \cdot w$
78. Find a unit vector that has the same direction as $v = 〈 - 6, - 2 〉$ $v = 〈 - 6, - 2 〉$ . [8.6]
79. Express the vector $t = 〈 - 9, 4 〉$ $t = 〈 - 9, 4 〉$ as a linear combination of the unit vectors i and j. [8.6]
80. Determine the direction angle $θ$ $θ$ of the vector $w = 〈 - 4, - 1 〉$ $w = 〈 - 4, - 1 〉$ to the nearest degree. [8.6]
81. Find the magnitude and the direction angle $θ$ $θ$ of $u = - 5 i - 3 j$ $u = - 5 i - 3 j$ . [8.6]
82. Find the angle between $u = 〈 3, - 7 〉$ $u = 〈 3, - 7 〉$ and $v = 〈 2, 2 〉$ $v = 〈 2, 2 〉$ to the nearest tenth of a degree. [8.6]
83. Airplane. An airplane has an airspeed of 160 mph. It is to make a flight in a direction of $80 °$ $80 °$ while there is a 20-mph wind from $310 °$ $310 °$ . What will the airplane’s actual heading be?[8.6]

Do the calculations in Exercises 84–87 for the vectors

$u = 2 i + 5 j$ $u = 2 i + 5 j$ , $v = - 3 i + 10 j$ $v = - 3 i + 10 j$ , and $w = 4 i + 7 j$ $w = 4 i + 7 j$ . [8.6]

84. $5 u - 8 v$ $5 u - 8 v$
85. $u - (v + w)$ $u - (v + w)$
86. $| u - v | $ $| u - v | $
87. $3 | w | + | v | $ $3 | w | + | v | $
88. Express the vector $\overset{⇀}{P Q}$ $\overset{⇀}{P Q}$ in the form $a i + b j$ $a i + b j$ , if P is the point $(1, - 3)$ $(1, - 3)$ and Q is the point $(- 4, 2)$ $(- 4, 2)$ . [8.6]

Express each vector in Exercises 89 and 90 in the form $a i + b j$ $a i + b j$ and sketch each in the coordinate plane. [8.6]

89. The unit vectors $u = (\cos θ) i + (\sin θ) j$ $u = (\cos θ) i + (\sin θ) j$ for $θ = π / 4$ $θ = π / 4$ and $θ = 5 π / 4$ $θ = 5 π / 4$ . Include the unit circle $x^{2} + y^{2} = 1$ $x^{2} + y^{2} = 1$ in your sketch.
90. The unit vector obtained by rotating j counterclockwise $2 π / 3$ $2 π / 3$ radians about the origin.
91. Express the vector $3 i - j$ $3 i - j$ as a product of its magnitude and its direction.
92. Which of the following is the trigonometric notation for $1 - i ?$ $1 - i ?$ [8.3]
1. $\sqrt{2} (\cos \frac{5 π}{4} + i \sin \frac{5 π}{4})$ $\sqrt{2} (\cos \frac{5 π}{4} + i \sin \frac{5 π}{4})$
2. $\sqrt{2} (\cos \frac{7 π}{4} - \sin \frac{7 π}{4})$ $\sqrt{2} (\cos \frac{7 π}{4} - \sin \frac{7 π}{4})$
3. $\cos \frac{7 π}{4} + i \sin \frac{7 π}{4}$ $\cos \frac{7 π}{4} + i \sin \frac{7 π}{4}$
4. $\sqrt{2} (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})$ $\sqrt{2} (\cos \frac{7 π}{4} + i \sin \frac{7 π}{4})$
93. Convert the polar equation $r = 100$ $r = 100$ to a rectangular equation. [8.4]
1. $x^{2} + y^{2} = 10,000$ $x^{2} + y^{2} = 10,000$
2. $x^{2} + y^{2} = 100$ $x^{2} + y^{2} = 100$
3. $\sqrt{x^{2} + y^{2}} = 10$ $\sqrt{x^{2} + y^{2}} = 10$
4. $\sqrt{x^{2} + y^{2}} = 1000$ $\sqrt{x^{2} + y^{2}} = 1000$
94. The graph of $r = 1 - 2 \cos θ$ $r = 1 - 2 \cos θ$ is which of the following?[8.4]

Synthesis

95. Let $u = 12 i + 5 j$ $u = 12 i + 5 j$ . Find a vector that has the same direction as u but has length 3. [8.6]
96. A parallelogram has sides of lengths 3.42 and 6.97. Its area is 18.4. Find the sizes of its angles. [8.1]

Collaborative Discussion and Writing

97. Summarize how you can tell algebraically when solving triangles whether there is no solution, one solution, or two solutions. [8.1], [8.2]
98. Give an example of an equation that is easier to graph in polar notation than in rectangular notation and explain why. [8.4]
99. Explain why the rectangular coordinates of a point are unique and the polar coordinates of a point are not unique. [8.4]
100. Explain why vectors $\overset{⇀}{Q R}$ $\overset{⇀}{Q R}$ and $\overset{⇀}{R Q}$ $\overset{⇀}{R Q}$ are not equivalent. [8.5]
101. Explain how unit vectors are related to the unit circle. [8.6]
102. Write a vector sum problem for a classmate for which the answer is $v = 5 i - 8 j$ $v = 5 i - 8 j$ . [8.6]

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Table of Contents for Review Exercises

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Table of Contents for
Review Exercises