1. Slide 7.4 - 1 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

2. OBJECTIVES Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Polar Coordinates Learn vocabulary for polar coordinates. Learn conversion between polar and rectangular coordinates. Learn to convert equations between rectangular and polar forms. Learn to graph polar equations. SECTION 7.4 1 2 3 4

3. Slide 7.4 - 3 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley POLAR COORDINATES In a polar coordinate system, we draw a horizontal ray in the plane. The ray is called the polar axis, and its endpoint is called the pole. A point P in the plane is described by an ordered pair of numbers (r,  ), and we refer to r and  as polar coordinates of P.

4. Slide 7.4 - 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley POLAR COORDINATES The point P(r,  ) in the polar coordinate system. r is the “directed distance” from the pole O to the point P.  is a directed angle from the polar axis to the line segment OP.

5. Slide 7.4 - 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley POLAR COORDINATES The polar coordinates of a point are not unique. The polar coordinates (3, 60º), (3, 420º), and (3, –300º) all represent the same point. In general, if a point P has polar coordinates (r,  ), then for any integer n, (r,  + n 360º) or (r,  + 2nπ) are also polar coordinates of P.

6. Slide 7.4 - 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Finding Different Polar Coordinates Plot the point P with polar coordinates (3, 225º). Find another pair of polar coordinates of P for which the following is true. a. r < 0 and 0º <  < 360º b. r < 0 and –360º <  < 0º c. r > 0 and –360º <  < 0º Solution

7. Slide 7.4 - 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued a. r < 0 and 0º <  < 360º EXAMPLE 1 Finding Different Polar Coordinates

8. Slide 7.4 - 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 1 Finding Different Polar Coordinates b. r < 0 and –360º <  < 0º

9. Slide 7.4 - 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 1 Finding Different Polar Coordinates c. r > 0 and –360º <  < 0º

10. Slide 7.4 - 10 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley POLAR AND RECTANGULAR COORDINATES Let the positive x-axis of the rectangular coordinate system serve as the polar axis and the origin as the pole for the polar coordinate system. Each point P has both polar coordinates (r,  ) and rectangular coordinates (x, y).

11. Slide 7.4 - 11 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley RELATIONSHIP BETWEEN POLAR AND RECTANGULAR COORDINATES

12. Slide 7.4 - 12 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CONVERTING FROM POLAR TO RECTANGULAR COORDINATES To convert the polar coordinates (r,  ) of a point to rectangular coordinates, use the equations

13. Slide 7.4 - 13 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Convert from Polar to Rectangular Coordinates Convert the polar coordinates of each point to rectangular coordinates. Solution The rectangular coordinates of (2, –30º) are

14. Slide 7.4 - 14 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Convert from Polar to Rectangular Coordinates Solution continued The rectangular coordinates of are

15. Slide 7.4 - 15 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CONVERTING FROM RECTANGULAR TO POLAR COORDINATES To convert the rectangular coordinates (x, y) of a point to polar coordinates, 1.Find the quadrant in which the given point (x, y) lies. 2.Useto find r. 3.Find  by usingand choose  so that it lies in the same quadrant as (x, y).

16. Slide 7.4 - 16 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Convert from Rectangular to Polar Coordinates Find polar coordinates (r,  ) of the point P whose rectangular coordinates are with r > 0 and 0 ≤  < 2π. 2.2. Solution 1. The point Plies in quadrant II with

17. Slide 7.4 - 17 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Convert from Rectangular to Polar Coordinates 3.3. Solution continued Choose because it lies in quadrant II. The polar coordinates of

18. Slide 7.4 - 18 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EQUATIONS IN RECTANGULAR AND POLAR FORMS An equation that has the rectangular coordinates x and y as variable is called a rectangular (or Cartesian) equation. An equation where the polar coordinates r and  are the variables is called a polar equation. Some examples of polar equations are

19. Slide 7.4 - 19 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CONVERTING AN EQUATION FROM RECTANGULAR TO POLAR FORM To convert a rectangular equation to a polar equation, we simply replace x by r cos  and y by r sin , and then simplify where possible.

20. Slide 7.4 - 20 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Converting an Equation from Rectangular to Polar Form Convert the equation x 2 + y 2 – 3x + 4 = 0 to polar form. Solution The equation of the rectangular equation x 2 + y 2 – 3x + 4 = 0. is the polar form

21. Slide 7.4 - 21 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CONVERTING AN EQUATION FROM POLAR TO RECTANGULAR FORM Converting an equation from polar to rectangular form will frequently require some ingenuity in order to use the substitutions

22. Slide 7.4 - 22 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Convert each polar equation to a rectangular equation and identify its graph. Solution Circle: center (0, 0) radius = 3

23. Slide 7.4 - 23 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Solution continued Line through the origin with a slope of 1.

24. Slide 7.4 - 24 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Solution continued Horizontal line with y-intercept = 1.

25. Slide 7.4 - 25 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 6 Converting an Equation from Polar to Rectangular Form Solution continued Circle: Center (1, 0) radius = 1

26. Slide 7.4 - 26 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley THE GRAPH OF A POLAR EQUATION To graph a polar equation we plot points in polar coordinates. The graph of a polar equation is the set of all points P(r,  ) that have at least one polar coordinate representation that satisfies the equation. Make a table of several ordered pair solutions (r,  ) of the equation, plot the points and join them with a smooth curve.

27. Slide 7.4 - 27 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Sketching the Graph of a Polar Equation Sketch the graph of the polar equation Solution cos (–  ) = cos , so the graph is symmetric in the polar axis, so compute values for 0 ≤  ≤ π.

28. Slide 7.4 - 28 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution continued EXAMPLE 7 Sketching the Graph of a Polar Equation This type of curve is called a cardiod because it resembles a heart.

29. Slide 7.4 - 29 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SYMMETRY IN POLAR EQUATIONS Symmetry with respect to the polar axis (x-axis) Replace (r,  ) by (r, –  ) or (–r, π –  ).

30. Slide 7.4 - 30 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SYMMETRY IN POLAR EQUATIONS Symmetry with respect to the line (y-axis) Replace (r,  ) by (r, π –  ) or (–r, –  ).

31. Slide 7.4 - 31 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SYMMETRY IN POLAR EQUATIONS Symmetry with respect to the pole Replace (r,  ) by (r, π +  ) or (–r,  ).

32. Slide 7.4 - 32 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley LIMAÇONS

33. Slide 7.4 - 33 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley LIMAÇONS

34. Slide 7.4 - 34 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ROSE CURVES If n is odd, the rose has n petals. If n is even, the rose has 2n petals

35. Slide 7.4 - 35 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley ROSE CURVES If n is odd, the rose has n petals. If n is even, the rose has 2n petals

36. Slide 7.4 - 36 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley CIRCLES

37. Slide 7.4 - 37 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley LEMNISCATES

38. Slide 7.4 - 38 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley SPIRALS