The document discusses polar equations and their use in describing curves. Polar equations are defined as equations involving the variables r and θ. Common polar equations like r = c, θ = c, r = ±c*cos(θ), and r = ±c*sin(θ) are presented along with examples of how they describe geometric shapes like circles and lines. In particular, the equations r = ±c*cos(θ) and r = ±c*sin(θ) always describe circles.

1. Polar Equations

2. Polar Equations Polars equations are equations in the variables r and .

3. Polar Equations Polars equations are equations in the variables r and . Many curves may be described easier using relations in r and  rather than relations between x and y.

4. Polar Equations Polars equations are equations in the variables r and . Many curves may be described easier using relations in r and  rather than relations between x and y.

5. Polar Equations Polars equations are equations in the variables r and . Many curves may be described easier using relations in r and  rather than relations between x and y.

6. Polar Equations The Constant Equations r = c &  =c  Frank Ma 2006

7. Polar Equations The Constant Equations r = c &  =c I. The equations r= c,

8. Polar Equations The Constant Equations r = c &  =c The equation r = c, distance from the point to the origin = c, and  any number.

9. Polar Equations The Constant Equations r = c &  =c The equation r = c, distance from the point to the origin = c, and  any number. This equation describes the circle of radius c, centered at (0,0).  Frank Ma 2006

10. Polar Equations The Constant Equations r = c &  =c The equation r = c, distance from the point to the origin = c, and  any number. This equation describes the circle of radius c, centered at (0,0).

11. Polar Equations The Constant Equations r = c &  =c II. The equation  = c,

12. Polar Equations The Constant Equations r = c &  =c II. The equation  = c, Directional angle to the point= c, and r any number.

13. Polar Equations The Constant Equations r = c &  =c II. The equation  = c, Directional angle to the point= c, and r any number. This equation describes the line making the angle c to x-axis.

14. Polar Equations The Constant Equations r = c &  =c II. The equation  = c, Directional angle to the point= c, and r any number. This equation describes the line making the angle c to x-axis. r>0  Frank Ma 2006 =C r<0

15. Polar Equations r = ±c*cos() & r = ±c*sin()

16. Polar Equations r = ±c*cos() & r = ±c*sin() The equations r = ±c*cos() r = ±c*sin() are circles.

17. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles.

18. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

19. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

20. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

21. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

22. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

23. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

24. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

25. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

26. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π  Frank Ma 2006

27. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

28. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

29. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

30. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

31. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

32. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

33. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

34. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

35. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

36. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() The equations r = ±c*cos() r = ±c*sin() are circles. r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

37. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

38. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

39. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

40. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

41. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

42. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

43. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

44. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π

45. Polar Equations r = ±c*cos() & r = ±c*sin() Example: Graph r = -sin() r  0 π ½ 7π/6 2/2 5π/4 3/2 4π/3 1 3π/2 3/2 5π/3 2/2 7π/4 ½ 11π/6 0 2π r  0 0 -½ π/6 -2/2 π/4 -3/2 π/3 -1 π/2 -3/2 2π/3 -2/2 3π/4 -½ 5π/6 0 π Remark: The graph consists of two circles as  goes from 0 to 2π

46. Polar Equations r = ±c*cos() & r = ±c*sin() The graphs of r = ±c*cos() & r = ±c*sin()consists of two overlapping circles tangent to the axes at the origin as  goes from 0 to 2π. r = c*sin() r = -c*cos() r = +c*cos() r = -c*sin()

47. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin()  Frank Ma 2006

48. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π

49. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π

50. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π  Frank Ma 2006

51. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π  Frank Ma 2006

52. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π

53. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π

54. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π

55. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π  Frank Ma 2006

56. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π  Frank Ma 2006

57. Polar Equations r = c(1 ± cos()) & r =c(1 ± sin()) Example: Graph r = 1 – sin() r  1 π 3/2 7π/6 1+2/2 5π/4 1+3/2 4π/3 2 3π/2 1+3/2 5π/3 1+2/2 7π/4 3/2 11π/6 1 2π r  1 0 ½ π/6 1-2/2 π/4 1-3/2 π/3 0 π/2 1-3/2 2π/3 1-2/2 3π/4 ½ 5π/6 1 π

58. Polar Equations r = cos(n) & r = c*sin(n)

59. Polar Equations r = cos(n) & r = c*sin(n) The following steps help us to graph polar equations, especially equations made up with sine and cosine of :  Frank Ma 2006

60. Polar Equations r = cos(n) & r = c*sin(n) The following steps help us to graph polar equations, especially equations made up with sine and cosine of : 1. Find 0o <  < 360o where r=0

61. Polar Equations r = cos(n) & r = c*sin(n) The following steps help us to graph polar equations, especially equations made up with sine and cosine of : 1. Find 0o <  < 360o where r=0 2. Find  between 0 and 360o where |r| is greatest.

62. Polar Equations r = cos(n) & r = c*sin(n) The following steps help us to graph polar equations, especially equations made up with sine and cosine of : 1. Find 0o <  < 360o where r=0 2. Find  between 0 and 360o where |r| is greatest. 3. Trace the curves using 1 and 2.  Frank Ma 2006

63. Polar Equations Example: r = sin(2)

64. Polar Equations Example: r = sin(2) Find r = 0 = sin(2),

65. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720.  Frank Ma 2006

66. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270

67. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270 Find r = 1 = sin(2),

68. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270 Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225.  Frank Ma 2006

69. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270 Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225. Find r = -1,  Frank Ma 2006

70. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270 Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225. Find r = -1, 2 = 270, 630,  = 135, 315

71. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270 Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225. Find r = -1, 2 = 270, 630,  = 135, 315 90 Draw the directions that r = 0. 180 0 1 270

72. Polar Equations Example: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270 Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225. Find r = -1, 2 = 270, 630,  = 135, 315 45 90 Draw the directions that r = 0. 135 Draw the directions that r = ±1.  Frank Ma 2006 180 0 1 225 315 270

73. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 45 90 135  Frank Ma 2006 180 0 1 225 315 270

74. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 0 <  < 90  0 < 2 < 180, sin(2) goes from 0 to 1 back to 0. 45 90 135  Frank Ma 2006 180 0 1 225 315 270

75. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 0 <  < 90  0 < 2 < 180, sin(2) goes from 0 to 1 back to 0. 45 90 135  Frank Ma 2006 180 0 1 225 315 270

76. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 0 <  < 90  0 < 2 < 180, sin(2) goes from 0 to 1 back to 0. 90 <  <180  180 < 2 < 360, sin(2) goes from 0 to -1 to 0. 45 90 135 180 0 1 225 315 270

77. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 0 <  < 90  0 < 2 < 180, sin(2) goes from 0 to 1 back to 0. 90 <  <180  180 < 2 < 360, sin(2) goes from 0 to -1 to 0. 45 90 135 180 0 1 225 315 270

78. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 0 <  < 90  0 < 2 < 180, sin(2) goes from 0 to 1 back to 0. 90 <  <180  180 < 2 < 360, sin(2) goes from 0 to -1 to 0. The similar observation about the other two sectors gives us the complete graph. 45 90 135  Frank Ma 2006 180 0 1 225 315 270

79. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 0 <  < 90  0 < 2 < 180, sin(2) goes from 0 to 1 back to 0. 90 <  <180  180 < 2 < 360, sin(2) goes from 0 to -1 to 0. The similar observation about the other two sectors gives us the complete graph. 45 90 135 180 0 1 225 315 270

80. Polar Equations Example: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 : 0 <  < 90  0 < 2 < 180, sin(2) goes from 0 to 1 back to 0. 90 <  <180  180 < 2 < 360, sin(2) goes from 0 to -1 to 0. The similar observation about the other two sectors gives us the complete graph. 45 90 135 This is known as the four-pedal-rose curve.  Frank Ma 2006 180 0 1 225 315 270

81. Equation Conversion

82. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form:  Frank Ma 2006

83. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2, r = x2 + y2 tan() = y/x  Frank Ma 2006

84. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y.  Frank Ma 2006

85. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y. r = 3 square both sides

86. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y. r = 3 square both sides r2 = 9

87. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y. r = 3 square both sides r2 = 9 replace into x&y  Frank Ma 2006

88. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y. r = 3 square both sides r2 = 9 replace into x&y x2 + y2 = 9  Frank Ma 2006

89. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y.

90. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y. r = 3 – 3cos()  Frank Ma 2006

91. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y. r = 3 – 3cos(), multiply by r

92. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y. r = 3 – 3cos(), multiply by r r2 = 3r – 3*r*cos()

93. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y. r = 3 – 3cos(), multiply by r r2 = 3r – 3*r*cos() in x&y  Frank Ma 2006

94. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y. r = 3 – 3cos(), multiply by r r2= 3r – 3*r*cos() in x&y x2 + y2 = 3x2 + y2 – 3x

95. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y. r = 3 – 3cos(), multiply by r r2 = 3r – 3*r*cos() in x&y x2 + y2 = 3x2 + y2 – 3x

96. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation.

97. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation. 2x2 = 3x – 2y2 – 8  Frank Ma 2006

98. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation. 2x2 = 3x – 2y2 – 8 2x2 + 2y2 = 3x – 8  Frank Ma 2006

99. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation. 2x2 = 3x – 2y2 – 8 2x2 + 2y2 = 3x – 8 2(x2 + y2) = 3x – 8

100. Equation Conversion Conversion Rule: To convert equations between the polar and rectangular form: x = r*cos() y = r*sin() r2 = x2 + y2 tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation. 2x2 = 3x – 2y2 – 8 2x2 + 2y2 = 3x – 8 2(x2 + y2) = 3x – 8 2r2 = 3rcos() – 8