1. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 8 Complex Numbers, Polar Equations, and Parametric Equations Copyright © 2013, 2009, 2005 Pearson Education, Inc. 45 Sections 8.5–8.6

2. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 46 8.5Polar Equations and Graphs Complex Numbers, Polar Equations, and Parametric Equations 8

3. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 47 Polar Equations and Graphs 8.5 Polar Coordinate System ▪ Graphs of Polar Equations ▪ Converting from Polar to Rectangular Equations ▪ Classifying Polar Equations

4. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 48 Plot each point by hand in the polar coordinate system. Then, determine the rectangular coordinates of each point. 8.5 Example 1 Plotting Points With Polar Coordinates (page 380) The rectangular coordinates of P(4, 135°) are

5. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 49 8.5 Example 1 Plotting Points With Polar Coordinates (cont.) The rectangular coordinates of

6. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 50 8.5 Example 1 Plotting Points With Polar Coordinates (cont.) The rectangular coordinates of are (0, –2).

7. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 51 Give three other pairs of polar coordinates for the point P(5, –110°). 8.5 Example 2(a) Giving Alternative Forms for Coordinates of a Point (page 381) Three pairs of polar coordinates for the point P(5, −110º) are (5, 250º), (−5, 70º), and (−5, −290º). Other answers are possible.

8. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 52 Give two pairs of polar coordinates for the point with the rectangular coordinates 8.5 Example 2(b) Giving Alternative Forms for Coordinates of a Point (page 394) The point lies in quadrant II. Since, one possible value for θ is 300°. Other answers are possible. Two pairs of polar coordinates are (12, 300°) and (−12, 120°).

9. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 53 8.5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles (page 382) For each rectangular equation, give the equivalent polar equation and sketch its graph. (a)y = 2x – 4 In standard form, the equation is 2x – y = 4, so a = 2, b = –1, and c = 4. The general form for the polar equation of a line is y = 2x – 4 is equivalent to

10. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 54 8.5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles (cont.)

11. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 55 8.5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles (cont.) This is the equation of a circle with center at the origin and radius 5. Note that in polar coordinates it is possible for r < 0.

12. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 56 8.5 Example 3 Examining Polar and Rectangular Equations of Lines and Circles (cont.)

13. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 57 8.5 Example 4 Graphing a Polar Equation (Cardioid) (page 383) Find some ordered pairs to determine a pattern of values of r.

14. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 58 8.5 Example 4 Graphing a Polar Equation (Cardioid) (cont.) Connect the points in order from (1, 0°) to (.5, 30°) to (.1, 60°) and so on.

15. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 59 8.5 Example 4 Graphing a Polar Equation (Cardioid) (cont.) Graphing calculator solution

16. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 60 8.5 Example 5 Graphing a Polar Equation (Rose) (page 384) Find some ordered pairs to determine a pattern of values of r.

17. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 61 8.5 Example 5 Graphing a Polar Equation (Rose) (cont.) Connect the points in order from (4, 0°) to (3.6, 10°) to (2.0, 20°) and so on.

18. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 62 8.5 Example 5 Graphing a Polar Equation (Rose) (cont.) Graphing calculator solution

19. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 63 8.5 Example 6 Graphing a Polar Equation (Lemniscate) (page 385) The graph only exists for [0°, 90°] and [180°, 270°] because sin 2θ must be positive. Find some ordered pairs to determine a pattern of values of r.

20. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 64 8.5 Example 6 Graphing a Polar Equation (Lemniscate) (cont.) 0 0 ±2.8±2.1 ±2.8 ±2.1 ±2.8

21. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 65 8.5 Example 6 Graphing a Polar Equation (Lemniscate) (cont.)

22. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 66 8.5 Example 6 Graphing a Polar Equation (Lemniscate) (cont.) Graphing calculator solution

23. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 67 8.5 Example 7 Graphing a Polar Equation (Spiral of Archimedes) (page 385) Graph r = –θ (θ measured in radians). Find some ordered pairs to determine a pattern of values of r.

24. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 68 8.5 Example 7 Graphing a Polar Equation (Spiral of Archimedes) (cont.)

25. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 69 8.5 Example 8 Converting a Polar Equation to a Rectangular Equation (page 386) Convert the equation to rectangular coordinates and graph. Multiply both sides by 1 – cos θ. Square both sides.

26. Copyright © 2013, 2009, 2005 Pearson Education, Inc. 70 8.5 Example 8 Converting a Polar Equation to a Rectangular Equation (cont.) The graph is a parabola with vertex at and axis y = 0.