1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter 6 Applications of Trigonometry

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.1 Vectors in the Plane

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 5 Quick Review

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 6 Quick Review Solutions

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7 Quick Review Solutions

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8 What you’ll learn about Two-Dimensional Vectors Vector Operations Unit Vectors Direction Angles Applications of Vectors … and why These topics are important in many real-world applications, such as calculating the effect of the wind on an airplane’s path.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9 Directed Line Segment

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 10 Two-Dimensional Vector

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 11 Initial Point, Terminal Point, Equivalent

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 12 Magnitude

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 13 Example Finding Magnitude of a Vector

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 14 Example Finding Magnitude of a Vector

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 15 Vector Addition and Scalar Multiplication

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 16 Example Performing Vector Operations

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 17 Example Performing Vector Operations

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 18 Unit Vectors

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 19 Example Finding a Unit Vector

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 20 Example Finding a Unit Vector

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 21 Standard Unit Vectors

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 22 Resolving the Vector

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 23 Example Finding the Components of a Vector

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 24 Example Finding the Components of a Vector

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 25 Example Finding the Direction Angle of a Vector

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 26 Example Finding the Direction Angle of a Vector

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 27 Velocity and Speed The velocity of a moving object is a vector because velocity has both magnitude and direction. The magnitude of velocity is speed.

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.2 Dot Product of Vectors

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 29 Quick Review

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 30 Quick Review Solutions

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 31 What you’ll learn about The Dot Product Angle Between Vectors Projecting One Vector onto Another Work … and why Vectors are used extensively in mathematics and science applications such as determining the net effect of several forces acting on an object and computing the work done by a force acting on an object.

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 32 Dot Product

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 33 Properties of the Dot Product Let u, v, and w be vectors and let c be a scalar. 1. u·v=v·u 2. u·u=|u| 2 3. 0·u=0 4. u·(v+w)=u·v+u·w (u+v) ·w=u·w+v·w 5. (cu) ·v=u·(cv)=c(u·v)

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 34 Example Finding the Dot Product

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 35 Example Finding the Dot Product

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 36 Angle Between Two Vectors

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 37 Example Finding the Angle Between Vectors

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 38 Example Finding the Angle Between Vectors

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 39 Orthogonal Vectors The vectors u and v are orthogonal if and only if u·v = 0.

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 40 Projection of u and v

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 41 Work

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.3 Parametric Equations and Motion

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 43 Quick Review

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 44 Quick Review Solutions

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 45 What you’ll learn about Parametric Equations Parametric Curves Eliminating the Parameter Lines and Line Segments Simulating Motion with a Grapher … and why These topics can be used to model the path of an object such as a baseball or golf ball.

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 46 Parametric Curve, Parametric Equations The graph of the ordered pairs (x,y) where x = f(t) and y = g(t) are functions defined on an interval I of t-values is a parametric curve. The equations are parametric equations for the curve, the variable t is a parameter, and I is the parameter interval.

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 47 Example Graphing Parametric Equations

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 48 Example Graphing Parametric Equations

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 49 Example Eliminating the Parameter

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 50 Example Eliminating the Parameter

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 51 Example Eliminating the Parameter

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 52 Example Eliminating the Parameter

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 53 Example Finding Parametric Equations for a Line

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 54 Example Finding Parametric Equations for a Line

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.4 Polar Coordinates

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 56 Quick Review

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 57 Quick Review Use the Law of Cosines to find the measure of the third side of the given triangle. 4. 40 º 8 10 5. 35 º 6 11

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 58 Quick Review Solutions

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 59 Quick Review Solutions Use the Law of Cosines to find the measure of the third side of the given triangle. 4. 40 º 8 10 5. 35 º 6 11 6.4 7

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 60 What you’ll learn about Polar Coordinate System Coordinate Conversion Equation Conversion Finding Distance Using Polar Coordinates … and why Use of polar coordinates sometimes simplifies complicated rectangular equations and they are useful in calculus.

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 61 The Polar Coordinate System

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 62 Example Plotting Points in the Polar Coordinate System

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 63 Example Plotting Points in the Polar Coordinate System

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 64 Finding all Polar Coordinates of a Point

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 65 Coordinate Conversion Equations

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 66 Example Converting from Polar to Rectangular Coordinates

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 67 Example Converting from Polar to Rectangular Coordinates

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 68 Example Converting from Rectangular to Polar Coordinates

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 69 Example Converting from Rectangular to Polar Coordinates

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 70 Example Converting from Polar Form to Rectangular Form

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 71 Example Converting from Polar Form to Rectangular Form

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 72 Example Converting from Polar Form to Rectangular Form

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 73 Example Converting from Polar Form to Rectangular Form

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.5 Graphs of Polar Equations

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 75 Quick Review

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 76 Quick Review Solutions

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 77 What you’ll learn about Polar Curves and Parametric Curves Symmetry Analyzing Polar Curves Rose Curves Limaçon Curves Other Polar Curves … and why Graphs that have circular or cylindrical symmetry often have simple polar equations, which is very useful in calculus.

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 78 Symmetry The three types of symmetry figures to be considered will have are: 1. The x-axis (polar axis) as a line of symmetry. 2. The y-axis (the line θ = π/2) as a line of symmetry. 3. The origin (the pole) as a point of symmetry.

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 79 Symmetry Tests for Polar Graphs The graph of a polar equation has the indicated symmetry if either replacement produces an equivalent polar equation. To Test for SymmetryReplaceBy 1. about the x-axis(r,θ) (r,-θ) or (-r, π-θ) 2. about the y-axis(r,θ) (-r,-θ) or (r, π-θ) 3. about the origin(r,θ) (-r,θ) or (r, π+θ)

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 80 Example Testing for Symmetry

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 81 Example Testing for Symmetry

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 82 Rose Curves

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 83 Limaçon Curves

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 6.6 De Moivre’s Theorem and nth Roots

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 85 Quick Review

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 86 Quick Review Solutions

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 87 What you’ll learn about The Complex Plane Trigonometric Form of Complex Numbers Multiplication and Division of Complex Numbers Powers of Complex Numbers Roots of Complex Numbers … and why The material extends your equation-solving technique to include equations of the form z n = c, n is an integer and c is a complex number.

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 88 Complex Plane

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 89 Absolute Value (Modulus) of a Complex Number

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 90 Graph of z = a + bi

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 91 Trigonometric Form of a Complex Number

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 92 Example Finding Trigonometric Form

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 93 Example Finding Trigonometric Form

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 94 Product and Quotient of Complex Numbers

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 95 Example Multiplying Complex Numbers

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 96 Example Multiplying Complex Numbers

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 97 A Geometric Interpretation of z 2

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 98 De Moivre’s Theorem

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 99 Example Using De Moivre’s Theorem

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 100 Example Using De Moivre’s Theorem

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 101 nth Root of a Complex Number

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 102 Finding nth Roots of a Complex Number

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 103 Example Finding Cube Roots

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 104 Example Finding Cube Roots

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 105 Chapter Test

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 106 Chapter Test

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 107 Chapter Test

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 108 Chapter Test Solutions

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 109 Chapter Test Solutions

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 110 Chapter Test Solutions