t7 polar equations and graphs

1. Polar Equations

2. Polar EquationsPolars equations are equations in the variables r and .

3. Polar EquationsPolars 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 EquationsThe Constant Equations r = c &  =c Frank Ma2006

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

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

9. Polar EquationsThe Constant Equations r = c &  =cThe 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 Ma2006

10. Polar EquationsThe Constant Equations r = c &  =cThe 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 EquationsThe Constant Equations r = c &  =cII. The equation  = c,

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

13. Polar EquationsThe Constant Equations r = c &  =cII. 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 EquationsThe Constant Equations r = c &  =cII. 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 Ma2006=Cr<0

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

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

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

18. Polar Equationsr = ±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/60 π

26. Polar Equationsr = ±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/60 π Frank Ma2006

37. Polar Equationsr = ±c*cos() & r = ±c*sin()Example: Graph r = -sin()r 0 π½ 7π/6 2/2 5π/43/2 4π/31 3π/23/2 5π/32/2 7π/4½ 11π/60 2πr 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/60 π

45. Polar Equationsr = ±c*cos() & r = ±c*sin()Example: Graph r = -sin()r 0 π½ 7π/6 2/2 5π/43/2 4π/31 3π/23/2 5π/32/2 7π/4½ 11π/60 2πr 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/60 πRemark: The graph consists of two circles as  goes from 0 to 2π

46. Polar Equationsr = ±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 Equationsr = c(1 ± cos()) & r =c(1 ± sin())Example: Graph r = 1 – sin() Frank Ma2006

48. Polar Equationsr = c(1 ± cos()) & r =c(1 ± sin())Example: Graph r = 1 – sin()r 1 π3/2 7π/6 1+2/2 5π/41+3/2 4π/32 3π/21+3/2 5π/31+2/2 7π/43/2 11π/61 2πr 1 0½ π/6 1-2/2 π/41-3/2 π/30 π/21-3/2 2π/31-2/2 3π/4½ 5π/61 π

50. Polar Equationsr = c(1 ± cos()) & r =c(1 ± sin())Example: Graph r = 1 – sin()r 1 π3/2 7π/6 1+2/2 5π/41+3/2 4π/32 3π/21+3/2 5π/31+2/2 7π/43/2 11π/61 2πr 1 0½ π/6 1-2/2 π/41-3/2 π/30 π/21-3/2 2π/31-2/2 3π/4½ 5π/61 π Frank Ma2006

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

59. Polar Equationsr = 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 Ma2006

60. Polar Equationsr = 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 Equationsr = 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=02. Find  between 0 and 360o where |r| is greatest.

62. Polar Equationsr = 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=02. Find  between 0 and 360o where |r| is greatest.3. Trace the curves using 1 and 2.  Frank Ma2006

63. Polar EquationsExample: r = sin(2)

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

65. Polar EquationsExample: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720. Frank Ma2006

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

68. Polar EquationsExample: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720.Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225. Frank Ma2006

69. Polar EquationsExample: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720.Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225.Find r = -1,  Frank Ma2006

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

71. Polar EquationsExample: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720.Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225.Find r = -1, 2 = 270, 630,  = 135, 31590Draw the directions that r = 0.18001270

72. Polar EquationsExample: r = sin(2) Find r = 0 = sin(2), 0 <  < 360 0 < 2 < 720.Therefore 2 = 0, 180, 360, 540, or  = 0, 90, 180, 270Find r = 1 = sin(2), 2 = 90, 450, or  = 45, 225.Find r = -1, 2 = 270, 630,  = 135, 3154590Draw the directions that r = 0.135Draw the directions that r = ±1. Frank Ma200618001225315270

73. Polar EquationsExample: r = sin(2) Investigate the graph in each sector from r = 0 to r = 0 :4590135 Frank Ma200618001225315270

74. Polar EquationsExample: 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. 4590135 Frank Ma200618001225315270

75. Polar EquationsExample: 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. 4590135 Frank Ma200618001225315270

76. Polar EquationsExample: 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. 459013518001225315270

77. Polar EquationsExample: 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. 459013518001225315270

78. Polar EquationsExample: 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.4590135 Frank Ma200618001225315270

79. Polar EquationsExample: 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.459013518001225315270

80. Polar EquationsExample: 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.4590135This is known as the four-pedal-rose curve. Frank Ma200618001225315270

81. Equation Conversion

82. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form: Frank Ma2006

83. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin()r2 = x2 + y2, r = x2 + y2tan() = y/x  Frank Ma2006

84. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y. Frank Ma2006

85. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y.r = 3 square both sides

86. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y.r = 3 square both sidesr2 = 9

87. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y.r = 3 square both sidesr2 = 9 replace into x&y Frank Ma2006

88. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 into rectangular equation in x&y.r = 3 square both sidesr2 = 9 replace into x&yx2 + y2 = 9 Frank Ma2006

89. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y.

90. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y.r = 3 – 3cos() Frank Ma2006

91. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the polar equation r = 3 – 3cos() into rectangular equation in x&y.r = 3 – 3cos(), multiply by r

92. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = 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 ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = 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 Ma2006

94. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = 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 ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = 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 ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation.

97. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation.2x2 = 3x – 2y2 – 8 Frank Ma2006

98. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation.2x2 = 3x – 2y2 – 82x2 + 2y2 = 3x – 8  Frank Ma2006

99. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation.2x2 = 3x – 2y2 – 82x2 + 2y2 = 3x – 8 2(x2 + y2) = 3x – 8

100. Equation ConversionConversion Rule:To convert equations between the polar and rectangular form:x = r*cos()y = r*sin() r2 = x2 + y2tan() = y/x Example: Convert the rectangular equation 2x2 = 3x – 2y2 – 8 into polar equation.2x2 = 3x – 2y2 – 82x2 + 2y2 = 3x – 8 2(x2 + y2) = 3x – 8 2r2 = 3rcos() – 8

t7 polar equations and graphs

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t7 polar equations and graphs