1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 1 Homework, Page 539 The polar coordinates of a point are given. Find its rectangular coordinates. 1.

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 2 Homework, Page 539 (a) Complete the table for the polar equation and (b) plot the corresponding points. 5.

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 3 Homework, Page 539 Plot the point with the given polar coordinates. 9.

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 4 Homework, Page 539 Plot the point with the given polar coordinates. 13.

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 5 Homework, Page 539 Find the rectangular coordinates of the point with the given polar coordinates. 17.

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 6 Homework, Page 539 Find the rectangular coordinates of the point with the given polar coordinates. 21.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 7 Homework, Page 539 Polar coordinates of point P are given. Find all of its polar coordinates. 25.

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 8 Homework, Page 539 Rectangular coordinates of point P are given. Find all polar coordinates of P that satisfy: (a) 0 ≤θ ≤2π (b) –π ≤ θ ≤ π (c) 0 ≤ θ ≤ 4π 29.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 9 Homework, Page 539 Match the polar equation with its graph. 33. c.

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 10 Homework, Page 539 Convert the polar equation to rectangular form and identify the graph. 37.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 11 Homework, Page 539 Convert the polar equation to rectangular form and identify the graph. 41.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 12 Homework, Page 539 Convert the rectangular equation to polar form and graph the polar equation. 45.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 13 Homework, Page 539 Convert the rectangular equation to polar form and graph the polar equation. 49.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 14 Homework, Page 539 53.A square with sides of length a and center at the origin has two sides parallel to the x-axis. Find polar coordinates of the vertices.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 15 Homework, Page 539 57.If r ≠ 0, which of the following polar coordinate pairs represents the same point as the point with polar coordinates a. b. c. d. e.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 16 Homework, Page 539 61.

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 17 Homework, Page 539

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 18 Homework, Page 539

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 19 Homework, Page 539 Use the results of # 61 to find the distance between the points with the given polar coordinates. 65.

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

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 21 Quick Review

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 22 Quick Review Solutions

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 23 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.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 24 Polar Curves and Parametric Curves Polar curves are, in reality, a special type of parametric curves, where, for all values of θ in some parameter interval that suffices to produce a complete graph (in many cases, 0 ≤ θ ≤ 2π).

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 25 Symmetry The three types of symmetry figures to be considered 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.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 26 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 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)

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 27 Example Testing for Symmetry

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 28 Rose Curves

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

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 30 Example Analyzing Polar Graphs

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 31 Example Analyzing Polar Graphs

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 32 Example Analyzing a Polar Graph Analyze the polar graph of Domain: Range: Continuity: Symmetry: Boundedness: Maximum r-value: Asymptotes:

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 33 Homework Homework Assignment #6 Read Section 6.6 Page 548, Exercises: 1 – 69 (EOO) Quiz next time

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

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 35 Quick Review

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 36 Quick Review Solutions

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 37 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.

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 38 Complex Plane

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 39 Addition of Complex Numbers

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 40 Absolute Value (Modulus) of a Complex Number

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 41 Graph of z = a + bi

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 42 Trigonometric Form of a Complex Number

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 43 Example Finding Trigonometric Form

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 44 Product and Quotient of Complex Numbers

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 45 Example Multiplying Complex Numbers

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 46 Example Dividing Complex Numbers

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 47 A Geometric Interpretation of z 2

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 48 De Moivre’s Theorem

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 49 Example Using De Moivre’s Theorem

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 50 nth Root of a Complex Number

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 51 Finding nth Roots of a Complex Number

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 6- 52 Example Finding Cube Roots