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Polar Coordinates
Polar Curves
Institute of Mathematics, University of the Philippines Diliman
Mathematics 54 (Elementary Analysis 2)
Polar Curves 1/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π/4)
2 B = (2,−π/4)
3 C = (−2, π/6)
4 D = (−3,−π/3)
Polar Curves 3/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π4
)2 B = (
2,−π4
) 3 C = (−2, π6)
4 D = (−3,−π3
)
Polar Curves 4/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π4
)= (1, 9π/4)
2 B = (2,−π
4
) 3 C = (−2, π6)
4 D = (−3,−π3
)
Polar Curves 4/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π4
)= (1, 9π/4) = (−1, 5π/4)
2 B = (2,−π
4
) 3 C = (−2, π6)
4 D = (−3,−π3
)
Polar Curves 4/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π4
)= (1, 9π/4) = (−1, 5π/4)
2 B = (2,−π
4
) = (2, 7π/4)
3 C = (−2, π6)
4 D = (−3,−π3
)
Polar Curves 4/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π4
)= (1, 9π/4) = (−1, 5π/4)
2 B = (2,−π
4
) = (2, 7π/4)
3 C = (−2, π6) = (2, 7π/6)
4 D = (−3,−π3
)
Polar Curves 4/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example.
Plot the following points:
1 A = (1, π4
)= (1, 9π/4) = (−1, 5π/4)
2 B = (2,−π
4
) = (2, 7π/4)
3 C = (−2, π6) = (2, 7π/6)
4 D = (−3,−π3
) = (3, 2π/3)
Polar Curves 4/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Conversion Equations
Polar to Cartesian
1 x = r cosθ
2 y = r sinθ
Cartesian to Polar
1 r2 = x2 +y2
2 tanθ = y
x
Polar Curves 5/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Conversion Equations
Polar to Cartesian
1 x = r cosθ
2 y = r sinθ
Cartesian to Polar
1 r2 = x2 +y2
2 tanθ = y
x
Polar Curves 5/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Conversion Equations
Polar to Cartesian
1 x = r cosθ
2 y = r sinθ
Cartesian to Polar
1 r2 = x2 +y2
2 tanθ = y
x
Polar Curves 5/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Conversion Equations
Polar to Cartesian
1 x = r cosθ
2 y = r sinθ
Cartesian to Polar
1 r2 = x2 +y2
2 tanθ = y
x
Polar Curves 5/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Conversion Equations
Polar to Cartesian
1 x = r cosθ
2 y = r sinθ
Cartesian to Polar
1 r2 = x2 +y2
2 tanθ = y
x
Polar Curves 5/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx .
Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12
=⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3
=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)
or(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 1.
Determine the polar coordinates of the point having Cartesian coordinates(−p3,1).
Solution. Recall that r2 = x2 +y2 and tanθ = yx . Thus,
r2 = (−p3)2 +12 =⇒ r = 2
tanθ = 1
−p3=⇒ θ = 5π
6
Hence, the polar coordinates are(2, 5π
6
)or
(−2, 11π
6
).
Polar Curves 6/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 2.
Determine the Cartesian coordinates of the point having polar coordinates(−5,−π3
).
Solution. Recall that x = r cosθ and y = r sinθ. Thus,
x =−5cos(−π
3
) =− 52
y =−5sin(−π
3
) = 5p
32
Hence, the Cartesian coordinates are(− 5
2 , 5p
32
).
Polar Curves 7/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 2.
Determine the Cartesian coordinates of the point having polar coordinates(−5,−π3
).
Solution. Recall that x = r cosθ and y = r sinθ.
Thus,
x =−5cos(−π
3
) =− 52
y =−5sin(−π
3
) = 5p
32
Hence, the Cartesian coordinates are(− 5
2 , 5p
32
).
Polar Curves 7/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 2.
Determine the Cartesian coordinates of the point having polar coordinates(−5,−π3
).
Solution. Recall that x = r cosθ and y = r sinθ. Thus,
x =−5cos(−π
3
)
=− 52
y =−5sin(−π
3
) = 5p
32
Hence, the Cartesian coordinates are(− 5
2 , 5p
32
).
Polar Curves 7/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 2.
Determine the Cartesian coordinates of the point having polar coordinates(−5,−π3
).
Solution. Recall that x = r cosθ and y = r sinθ. Thus,
x =−5cos(−π
3
) =− 52
y =−5sin(−π
3
) = 5p
32
Hence, the Cartesian coordinates are(− 5
2 , 5p
32
).
Polar Curves 7/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 2.
Determine the Cartesian coordinates of the point having polar coordinates(−5,−π3
).
Solution. Recall that x = r cosθ and y = r sinθ. Thus,
x =−5cos(−π
3
) =− 52
y =−5sin(−π
3
)
= 5p
32
Hence, the Cartesian coordinates are(− 5
2 , 5p
32
).
Polar Curves 7/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 2.
Determine the Cartesian coordinates of the point having polar coordinates(−5,−π3
).
Solution. Recall that x = r cosθ and y = r sinθ. Thus,
x =−5cos(−π
3
) =− 52
y =−5sin(−π
3
) = 5p
32
Hence, the Cartesian coordinates are(− 5
2 , 5p
32
).
Polar Curves 7/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Coordinates
Example 2.
Determine the Cartesian coordinates of the point having polar coordinates(−5,−π3
).
Solution. Recall that x = r cosθ and y = r sinθ. Thus,
x =−5cos(−π
3
) =− 52
y =−5sin(−π
3
) = 5p
32
Hence, the Cartesian coordinates are(− 5
2 , 5p
32
).
Polar Curves 7/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 1.
Sketch r = 2.
Remark. In general, the graph of the equation r = k is a circle centered at the poleof radius |k|. Note that r = k and r =−k represent the same curve.
Polar Curves 8/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 1.
Sketch r = 2.
Remark. In general, the graph of the equation r = k is a circle centered at the poleof radius |k|. Note that r = k and r =−k represent the same curve.
Polar Curves 8/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 1.
Sketch r = 2.
Remark. In general, the graph of the equation r = k is a circle centered at the poleof radius |k|.
Note that r = k and r =−k represent the same curve.
Polar Curves 8/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 1.
Sketch r = 2.
Remark. In general, the graph of the equation r = k is a circle centered at the poleof radius |k|. Note that r = k and r =−k represent the same curve.
Polar Curves 8/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 2.
Sketch θ = π
4.
Remark. In general, the graph of the equation θ = k is a line passing through thepole making an angle k with the polar axis. Also, its Cartesian form is y = (tank)x,when non-vertical, or x = 0, when vertical.
Polar Curves 9/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 2.
Sketch θ = π
4.
Remark. In general, the graph of the equation θ = k is a line passing through thepole making an angle k with the polar axis. Also, its Cartesian form is y = (tank)x,when non-vertical, or x = 0, when vertical.
Polar Curves 9/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 2.
Sketch θ = π
4.
Remark. In general, the graph of the equation θ = k is a line passing through thepole making an angle k with the polar axis.
Also, its Cartesian form is y = (tank)x,when non-vertical, or x = 0, when vertical.
Polar Curves 9/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 2.
Sketch θ = π
4.
Remark. In general, the graph of the equation θ = k is a line passing through thepole making an angle k with the polar axis. Also, its Cartesian form is y = (tank)x,when non-vertical,
or x = 0, when vertical.
Polar Curves 9/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 2.
Sketch θ = π
4.
Remark. In general, the graph of the equation θ = k is a line passing through thepole making an angle k with the polar axis. Also, its Cartesian form is y = (tank)x,when non-vertical, or x = 0, when vertical.
Polar Curves 9/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole.
We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve.
Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r = 4cosθ.
Remark. In general, the graph of the equation r = acosθ is a circle of radius |a/2|tangent to the line θ =π/2 at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 10/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole.
We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve.
Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Equations and Polar Curves
Example 3.
Sketch r =−5sinθ.
Remark. In general, the graph of the equation r = asinθ is a circle of radius |a/2|tangent to the polar axis at the pole. We only need to vary θ on [0,π] to trace outthe curve. Exercise: Find its Cartesian form.
Polar Curves 11/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Region
Illustration.
Let R be the set of points satisfying the conditions
1 É r É 2π
6É θ É π
3.
Polar Curves 12/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Region
Illustration.
Let R be the set of points satisfying the conditions
1 É r É 2π
6É θ É π
3.
Polar Curves 12/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Region
Illustration.
Let R be the set of points satisfying the conditions
1 É r É 2π
6É θ É π
3.
Polar Curves 12/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Region
Illustration.
Let R be the set of points satisfying the conditions
1 É r É 2π
6É θ É π
3.
Polar Curves 12/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Region
Illustration.
Let R be the set of points satisfying the conditions
1 É r É 2π
6É θ É π
3.
Polar Curves 12/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Region
Illustration.
Let R be the set of points satisfying the conditions
1 É r É 2π
6É θ É π
3.
Polar Curves 12/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Polar Regions
Exercises.
Graph the following set of points:
1 1 É |r| É 2, 2π/3 É θ É 5π/6
2 4 É r É 5
3 π/3 É θ É 2π/3
Exercises.
Find the polar equivalent of the following:
1 x = 2
2 xy = 1
3 x2 + (y−3)2 = 9
4 x = e2t cos t,y = e2t sin t, t ∈RFind the Cartesian form of the following:
1 r2 = 4r cosθ
2 r = 4
2cosθ− sinθ
Polar Curves 13/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = 0
A polar curve is symmetric about the line θ = 0 (or x−axis)
if whenever (r,θ), in itsequation, is replaced by (r,−θ) or by (−r,π−θ), equivalent equation is obtained.
Polar Curves 14/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = 0
A polar curve is symmetric about the line θ = 0 (or x−axis)
if whenever (r,θ), in itsequation, is replaced by (r,−θ) or by (−r,π−θ), equivalent equation is obtained.
Polar Curves 14/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = 0
A polar curve is symmetric about the line θ = 0 (or x−axis)
if whenever (r,θ), in itsequation, is replaced by (r,−θ) or by (−r,π−θ), equivalent equation is obtained.
Polar Curves 14/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = 0
A polar curve is symmetric about the line θ = 0 (or x−axis)
if whenever (r,θ), in itsequation, is replaced by (r,−θ) or by (−r,π−θ), equivalent equation is obtained.
Polar Curves 14/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = 0
A polar curve is symmetric about the line θ = 0 (or x−axis)
if whenever (r,θ), in itsequation, is replaced by (r,−θ) or by (−r,π−θ), equivalent equation is obtained.
Polar Curves 14/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = 0
A polar curve is symmetric about the line θ = 0 (or x−axis) if whenever (r,θ), in itsequation, is replaced by (r,−θ) or by (−r,π−θ), equivalent equation is obtained.
Polar Curves 14/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = π2
A polar curve is symmetric about the line θ = π2 (or y−axis)
if whenever (r,θ), in itsequation, is replaced by (r,π−θ) or by (−r,−θ), equivalent equation is obtained.
Polar Curves 15/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = π2
A polar curve is symmetric about the line θ = π2 (or y−axis)
if whenever (r,θ), in itsequation, is replaced by (r,π−θ) or by (−r,−θ), equivalent equation is obtained.
Polar Curves 15/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = π2
A polar curve is symmetric about the line θ = π2 (or y−axis)
if whenever (r,θ), in itsequation, is replaced by (r,π−θ) or by (−r,−θ), equivalent equation is obtained.
Polar Curves 15/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = π2
A polar curve is symmetric about the line θ = π2 (or y−axis)
if whenever (r,θ), in itsequation, is replaced by (r,π−θ) or by (−r,−θ), equivalent equation is obtained.
Polar Curves 15/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = π2
A polar curve is symmetric about the line θ = π2 (or y−axis)
if whenever (r,θ), in itsequation, is replaced by (r,π−θ) or by (−r,−θ), equivalent equation is obtained.
Polar Curves 15/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About θ = π2
A polar curve is symmetric about the line θ = π2 (or y−axis) if whenever (r,θ), in its
equation, is replaced by (r,π−θ) or by (−r,−θ), equivalent equation is obtained.
Polar Curves 15/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About the Pole
A polar curve is symmetric about the pole
if whenever (r,θ), in its equation, isreplaced by (−r,θ) or by (r,θ+π), an equivalent equation is obtained.
Polar Curves 16/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About the Pole
A polar curve is symmetric about the pole
if whenever (r,θ), in its equation, isreplaced by (−r,θ) or by (r,θ+π), an equivalent equation is obtained.
Polar Curves 16/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About the Pole
A polar curve is symmetric about the pole
if whenever (r,θ), in its equation, isreplaced by (−r,θ) or by (r,θ+π), an equivalent equation is obtained.
Polar Curves 16/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About the Pole
A polar curve is symmetric about the pole
if whenever (r,θ), in its equation, isreplaced by (−r,θ) or by (r,θ+π), an equivalent equation is obtained.
Polar Curves 16/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About the Pole
A polar curve is symmetric about the pole
if whenever (r,θ), in its equation, isreplaced by (−r,θ) or by (r,θ+π), an equivalent equation is obtained.
Polar Curves 16/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Symmetry in the Polar Plane
Symmetry About the Pole
A polar curve is symmetric about the pole if whenever (r,θ), in its equation, isreplaced by (−r,θ) or by (r,θ+π), an equivalent equation is obtained.
Polar Curves 16/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθ
r = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ)
=⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθ
r = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ)
=⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Limaçons
Limaçons are curves whose equations are of the form
r = a±bcosθ or
r = a±bsinθ where a,b > 0
Testing for symmetry,
r = a±bcosθr = a±bcos(−θ) =⇒ r = a±bcosθ
thus, symmetric with respect to the x−axis
r = a±bsinθr = a±bsin(π−θ) =⇒ r = a±bsinθ
thus, symmetric with respect to the y−axis
Polar Curves 17/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.
The type of limaçon depends on the ratio ab . Here, it’s a
b = 12 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+2cosθ.
The graph is called a limaçon with a loop.The type of limaçon depends on the ratio a
b . Here, it’s ab = 1
2 .
Polar Curves 18/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid.
Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 1+cosθ.
The graph is called a cardioid. Note that ab = 1.
Polar Curves 19/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent.
Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 3+2cosθ.
The graph is called a limaçon with a dent. Note that ab = 3
2 .
Polar Curves 20/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon.
Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Example.
Sketch r = 2+cosθ.
The graph is called a convex limaçon. Note that ab = 2.
Polar Curves 21/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Types of Limaçons
In summary, for r = a±bcosθ, where a,b > 0, we have
i.) 0 < ab < 1 limaçon with a loop
ii.) ab = 1 cardioid
iii.) 1 < ab < 2 limaçon with a dent
iv.) 2 É ab convex limaçon
Remark.
The graph of r =−a±bcosθ is the same as the graph of r = a±bcosθ
Polar Curves 22/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Types of Limaçons
In summary, for r = a±bcosθ, where a,b > 0, we have
i.) 0 < ab < 1 limaçon with a loop
ii.) ab = 1 cardioid
iii.) 1 < ab < 2 limaçon with a dent
iv.) 2 É ab convex limaçon
Remark.
The graph of r =−a±bcosθ is the same as the graph of r = a±bcosθ
Polar Curves 22/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Types of Limaçons
In summary, for r = a±bcosθ, where a,b > 0, we have
i.) 0 < ab < 1 limaçon with a loop
ii.) ab = 1 cardioid
iii.) 1 < ab < 2 limaçon with a dent
iv.) 2 É ab convex limaçon
Remark.
The graph of r =−a±bcosθ is the same as the graph of r = a±bcosθ
Polar Curves 22/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Types of Limaçons
In summary, for r = a±bcosθ, where a,b > 0, we have
i.) 0 < ab < 1 limaçon with a loop
ii.) ab = 1 cardioid
iii.) 1 < ab < 2 limaçon with a dent
iv.) 2 É ab convex limaçon
Remark.
The graph of r =−a±bcosθ is the same as the graph of r = a±bcosθ
Polar Curves 22/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Types of Limaçons
In summary, for r = a±bcosθ, where a,b > 0, we have
i.) 0 < ab < 1 limaçon with a loop
ii.) ab = 1 cardioid
iii.) 1 < ab < 2 limaçon with a dent
iv.) 2 É ab convex limaçon
Remark.
The graph of r =−a±bcosθ is the same as the graph of r = a±bcosθ
Polar Curves 22/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Types of Limaçons
In summary, for r = a±bcosθ, where a,b > 0, we have
i.) 0 < ab < 1 limaçon with a loop
ii.) ab = 1 cardioid
iii.) 1 < ab < 2 limaçon with a dent
iv.) 2 É ab convex limaçon
Remark.
The graph of r =−a±bcosθ is the same as the graph of r = a±bcosθ
Polar Curves 22/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
Types of Limaçons
In summary, for r = a±bcosθ, where a,b > 0, we have
i.) 0 < ab < 1 limaçon with a loop
ii.) ab = 1 cardioid
iii.) 1 < ab < 2 limaçon with a dent
iv.) 2 É ab convex limaçon
Remark.
The graph of r =−a±bcosθ is the same as the graph of r = a±bcosθ
Polar Curves 22/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
The graph of r = a±bcosθ is a limaçon oriented horizontally, i.e. symmetric alongx−axis.
r = a+bcosθ r = a−bcosθ
Polar Curves 23/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Limaçons
The graph of r = a±bsinθ is a limaçon oriented vertically, i.e. symmetric alongy−axis.
r = a+bsinθ r = a−bsinθ
Polar Curves 24/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ;
or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθ
r = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ))
=⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθ
thus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.
additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ))
=⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ
=⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθ
thus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.
additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Roses
Roses are curves whose equations are of the form
r = acosnθ; or
r = asinnθ where a > 0, n ∈N
Testing for symmetry,
r = acosnθr = acos(n(−θ)) =⇒ r = acosnθthus, symmetric along the x−axis.additionally, symmetric along y−axis for an even n
r = asinnθ−r = asin(n(−θ)) =⇒ −r =−asinnθ =⇒ r = asinnθthus, symmetric along the y−axis.additionally, symmetric along x−axis for an even n
Polar Curves 25/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.
In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2cos2θ.
The graph is a rose with 4 petals.In fact, the number of petals is 2n if n is even. And it’s n if n is odd.
Polar Curves 26/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.
Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Roses
Example.
Sketch the graph of r = 2sin3θ.
The graph is a rose with 3 petals.Here, n = 3 is odd. Hence, n = 3 is the number of petals.
Polar Curves 27/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ
r = 2sin4θ
Polar Curves 28/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ
r = 2sin4θ
Polar Curves 28/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ r = 2sin4θ
Polar Curves 28/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ r = 2sin4θ
Polar Curves 28/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ
r = 2sin9θ
Polar Curves 29/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ
r = 2sin9θ
Polar Curves 29/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ r = 2sin9θ
Polar Curves 29/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ r = 2sin9θ
Polar Curves 29/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
Exercises
Graph the following:
1 r = 2cosθ
2 r =−3cos2θ
3 r = sin4θ
4 r = 5cos5θ
Polar Curves 30/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
The graph of r2 = 6cos2θ is a lemniscate.
Polar Curves 31/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
The graph of r2 = 6cos2θ is a lemniscate.
Polar Curves 31/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
The graph of r2 = 6cos2θ is a lemniscate.
Polar Curves 31/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
The graph of r2 = 6cos2θ is a lemniscate.
Polar Curves 31/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
The graph of r2 = 6cos2θ is a lemniscate.
Polar Curves 31/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
The graph of r = θ,θ Ê 0 is the Archimedian spiral.
Polar Curves 32/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = 1+4cos5θ
Polar Curves 33/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = sin
(8θ
5
)
Polar Curves 34/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = esinθ −2cos4θ
Polar Curves 35/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = sin2(2.4θ)+cos4(2θ)
Polar Curves 36/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = sin2(1.2θ)+cos3(6θ)
Polar Curves 37/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = ecosθ −2cos4θ+ sin3(θ
3
)
Polar Curves 38/ 39
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
The "cannabis" curver =
(1+ 9
10 cos8θ)(
1+ 110 cos24θ
)(9
10 + 110 cos200θ
)(1+ sinθ)
Polar Curves 39/ 39