2. Applications of Trigonometry
Chapter 6
Applications of Trigonometry

3. 6.1
Vectors in the Plane

5. Quick Review

6. Quick Review

7. Quick Review

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. Directed Line Segment

10. Two-Dimensional Vector

11. Initial Point, Terminal Point, Equivalent

12. Magnitude

13. Example Finding Magnitude of a Vector

14. Vector Addition and Scalar Multiplication

15. Example Performing Vector Operations

16. Unit Vectors

17. Example Finding a Unit Vector

18. Standard Unit Vectors

19. Resolving the Vector

20. Example Finding the Components of a Vector

21. Example Finding the Direction Angle of a Vector

22. 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.

23. 6.2
Dot Product of Vectors

24. Quick Review

25. 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.

26. Dot Product

27. Properties of the Dot Product
Let u, v, and w be vectors and let c be a scalar.
u · v = v · u
u · u = |u|2
0 · u=0
u·(v+w)=u·v+u·w (u+v) ·w=u·w+v·w
(cu) ·v=u·(cv)=c(u·v)

28. Example Finding the Dot Product

29. Angle Between Two Vectors

30. Example Finding the Angle Between Vectors

31. Orthogonal Vectors
The vectors u and v are orthogonal if and only if u·v = 0.

32. Projection of u and v

33. Work

34. Parametric Equations and Motion
6.3
Parametric Equations and Motion

35. Quick Review Solutions

36. 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.

37. 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.

38. Example Graphing Parametric Equations

39. Example Graphing Parametric Equations

40. Example Eliminating the Parameter

41. Example Eliminating the Parameter

42. Example Finding Parametric Equations for a Line

43. 6.4
Polar Coordinates

44. Quick Review

45. Quick Review
Use the Law of Cosines to find the measure of the third
side of the given triangle. 4.
40º 8 10 5.
35º 6 11
6.4 7

46. 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.

47. The Polar Coordinate System

48. Example Plotting Points in the Polar Coordinate System

49. Finding all Polar Coordinates of a Point

50. Coordinate Conversion Equations

51. Example Converting from Polar to Rectangular Coordinates

52. Example Converting from Rectangular to Polar Coordinates

53. Example Converting from Polar Form to Rectangular Form

54. Example Converting from Polar Form to Rectangular Form

55. Graphs of Polar Equations
6.5
Graphs of Polar Equations

56. Quick Review

57. 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.

58. Symmetry
The three types of symmetry figures to be considered will have are:
The x-axis (polar axis) as a line of symmetry.
The y-axis (the line θ = π/2) as a line of symmetry.
The origin (the pole) as a point of symmetry.

59. 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 Symmetry Replace By
about the x-axis (r,θ) (r,-θ) or (-r, π-θ)
about the y-axis (r,θ) (-r,-θ) or (r, π-θ)
about the origin (r,θ) (-r,θ) or (r, π+θ)

60. Example Testing for Symmetry

61. Rose Curves

62. Limaçon Curves

63. De Moivre’s Theorem and nth Roots
6.6
De Moivre’s Theorem and nth Roots

64. Quick Review

65. 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 zn = c, n is an integer and c is a complex
number.

66. Complex Plane

67. Absolute Value (Modulus) of a Complex Number

68. Graph of z = a + bi

69. Trigonometric Form of a Complex Number

70. Example Finding Trigonometric Form

71. Product and Quotient of Complex Numbers

72. Example Multiplying Complex Numbers

73. A Geometric Interpretation of z2

74. De Moivre’s Theorem

75. Example Using De Moivre’s Theorem

76. nth Root of a Complex Number

77. Finding nth Roots of a Complex Number

78. Example Finding Cube Roots

79. Chapter Test

80. Chapter Test

81. Chapter Test