lecture 9 (polar coordinates and polar curves)
TRANSCRIPT
![Page 1: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/1.jpg)
Polar Coordinates
Polar Curves
Institute of Mathematics, University of the Philippines Diliman
Mathematics 54 (Elementary Analysis 2)
Polar Curves 1/ 39
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Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 3: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/3.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 4: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/4.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 5: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/5.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 6: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/6.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 7: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/7.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 8: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/8.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 9: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/9.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 10: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/10.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 11: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/11.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 12: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/12.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
The Polar Coordinate System
Polar Curves 2/ 39
![Page 13: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/13.jpg)
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
![Page 14: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/14.jpg)
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
![Page 15: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/15.jpg)
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
![Page 16: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/16.jpg)
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
![Page 17: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/17.jpg)
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
![Page 18: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/18.jpg)
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
![Page 19: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/19.jpg)
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
![Page 20: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/20.jpg)
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
![Page 21: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/21.jpg)
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
![Page 22: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/22.jpg)
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
![Page 23: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/23.jpg)
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
![Page 24: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/24.jpg)
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
![Page 25: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/25.jpg)
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
![Page 26: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/26.jpg)
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
![Page 27: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/27.jpg)
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
![Page 28: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/28.jpg)
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
![Page 29: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/29.jpg)
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
![Page 30: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/30.jpg)
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
![Page 31: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/31.jpg)
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
![Page 32: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/32.jpg)
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
![Page 33: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/33.jpg)
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
![Page 34: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/34.jpg)
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
![Page 35: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/35.jpg)
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
![Page 36: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/36.jpg)
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
![Page 37: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/37.jpg)
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
![Page 38: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/38.jpg)
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
![Page 39: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/39.jpg)
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
![Page 40: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/40.jpg)
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
![Page 41: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/41.jpg)
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
![Page 42: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/42.jpg)
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
![Page 43: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/43.jpg)
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
![Page 44: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/44.jpg)
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
![Page 45: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/45.jpg)
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
![Page 46: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/46.jpg)
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
![Page 47: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/47.jpg)
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
![Page 48: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/48.jpg)
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
![Page 49: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/49.jpg)
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
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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
![Page 51: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/51.jpg)
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
![Page 52: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/52.jpg)
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
![Page 53: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/53.jpg)
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
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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
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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
![Page 56: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/56.jpg)
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
![Page 57: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/57.jpg)
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
![Page 58: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/58.jpg)
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
![Page 94: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/94.jpg)
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
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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
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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
![Page 97: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/97.jpg)
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
![Page 98: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/98.jpg)
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
![Page 99: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/99.jpg)
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
![Page 100: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/100.jpg)
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
![Page 101: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/101.jpg)
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
![Page 102: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/102.jpg)
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
![Page 103: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/103.jpg)
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
![Page 104: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/104.jpg)
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
![Page 105: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/105.jpg)
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
![Page 106: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/106.jpg)
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
![Page 107: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/107.jpg)
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
![Page 108: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/108.jpg)
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
![Page 109: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/109.jpg)
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
![Page 110: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/110.jpg)
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
![Page 111: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/111.jpg)
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
![Page 112: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/112.jpg)
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
![Page 113: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/113.jpg)
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
![Page 114: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/114.jpg)
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
![Page 115: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/115.jpg)
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
![Page 116: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/116.jpg)
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
![Page 117: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/117.jpg)
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
![Page 118: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/118.jpg)
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
![Page 119: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/119.jpg)
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
![Page 120: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/120.jpg)
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
![Page 121: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/121.jpg)
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
![Page 122: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/122.jpg)
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
![Page 123: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/123.jpg)
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
![Page 124: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/124.jpg)
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
![Page 125: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/125.jpg)
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
![Page 126: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/126.jpg)
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
![Page 127: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/127.jpg)
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
![Page 128: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/128.jpg)
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
![Page 129: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/129.jpg)
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
![Page 130: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/130.jpg)
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
![Page 131: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/131.jpg)
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
![Page 132: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/132.jpg)
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
![Page 133: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/133.jpg)
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
![Page 134: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/134.jpg)
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
![Page 135: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/135.jpg)
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
![Page 136: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/136.jpg)
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
![Page 137: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/137.jpg)
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
![Page 138: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/138.jpg)
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
![Page 139: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/139.jpg)
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
![Page 140: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/140.jpg)
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
![Page 141: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/141.jpg)
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
![Page 142: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/142.jpg)
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
![Page 143: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/143.jpg)
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
![Page 144: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/144.jpg)
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
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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
![Page 146: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/146.jpg)
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
![Page 147: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/147.jpg)
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
![Page 148: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/148.jpg)
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
![Page 149: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/149.jpg)
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
![Page 150: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/150.jpg)
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
![Page 151: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/151.jpg)
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
![Page 152: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/152.jpg)
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
![Page 153: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/153.jpg)
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
![Page 154: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/154.jpg)
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
![Page 155: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/155.jpg)
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
![Page 156: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/156.jpg)
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
![Page 157: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/157.jpg)
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
![Page 158: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/158.jpg)
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
![Page 159: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/159.jpg)
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
![Page 160: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/160.jpg)
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
![Page 161: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/161.jpg)
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
![Page 162: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/162.jpg)
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
![Page 163: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/163.jpg)
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
![Page 164: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/164.jpg)
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
![Page 165: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/165.jpg)
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
![Page 166: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/166.jpg)
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
![Page 167: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/167.jpg)
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
![Page 168: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/168.jpg)
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
![Page 169: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/169.jpg)
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
![Page 170: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/170.jpg)
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
![Page 171: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/171.jpg)
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
![Page 172: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/172.jpg)
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
![Page 173: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/173.jpg)
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
![Page 174: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/174.jpg)
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
![Page 175: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/175.jpg)
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
![Page 176: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/176.jpg)
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
![Page 177: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/177.jpg)
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
![Page 178: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/178.jpg)
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
![Page 179: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/179.jpg)
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
![Page 180: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/180.jpg)
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
![Page 181: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/181.jpg)
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
![Page 182: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/182.jpg)
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
![Page 183: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/183.jpg)
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
![Page 184: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/184.jpg)
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
![Page 185: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/185.jpg)
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
![Page 186: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/186.jpg)
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
![Page 187: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/187.jpg)
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
![Page 188: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/188.jpg)
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
![Page 189: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/189.jpg)
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
![Page 190: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/190.jpg)
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
![Page 191: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/191.jpg)
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
![Page 192: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/192.jpg)
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
![Page 193: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/193.jpg)
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
![Page 194: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/194.jpg)
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
![Page 195: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/195.jpg)
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
![Page 196: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/196.jpg)
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
![Page 197: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/197.jpg)
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
![Page 198: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/198.jpg)
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
![Page 199: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/199.jpg)
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
![Page 200: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/200.jpg)
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
![Page 201: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/201.jpg)
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
![Page 202: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/202.jpg)
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
![Page 203: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/203.jpg)
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
![Page 204: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/204.jpg)
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
![Page 205: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/205.jpg)
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
![Page 206: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/206.jpg)
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
![Page 207: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/207.jpg)
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
![Page 208: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/208.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ
r = 2sin4θ
Polar Curves 28/ 39
![Page 209: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/209.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ
r = 2sin4θ
Polar Curves 28/ 39
![Page 210: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/210.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ r = 2sin4θ
Polar Curves 28/ 39
![Page 211: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/211.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos4θ r = 2sin4θ
Polar Curves 28/ 39
![Page 212: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/212.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ
r = 2sin9θ
Polar Curves 29/ 39
![Page 213: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/213.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ
r = 2sin9θ
Polar Curves 29/ 39
![Page 214: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/214.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ r = 2sin9θ
Polar Curves 29/ 39
![Page 215: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/215.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Special Curves
r = 2cos9θ r = 2sin9θ
Polar Curves 29/ 39
![Page 216: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/216.jpg)
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
![Page 217: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/217.jpg)
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
![Page 218: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/218.jpg)
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
![Page 219: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/219.jpg)
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
![Page 220: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/220.jpg)
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
![Page 221: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/221.jpg)
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
![Page 222: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/222.jpg)
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
![Page 223: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/223.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = 1+4cos5θ
Polar Curves 33/ 39
![Page 224: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/224.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = sin
(8θ
5
)
Polar Curves 34/ 39
![Page 225: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/225.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = esinθ −2cos4θ
Polar Curves 35/ 39
![Page 226: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/226.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = sin2(2.4θ)+cos4(2θ)
Polar Curves 36/ 39
![Page 227: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/227.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = sin2(1.2θ)+cos3(6θ)
Polar Curves 37/ 39
![Page 228: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/228.jpg)
Polar Coordinates Graphs in Polar Coordinates Special Curves in Polar Coordinates
Interesting Curves
Example.
r = ecosθ −2cos4θ+ sin3(θ
3
)
Polar Curves 38/ 39
![Page 229: Lecture 9 (Polar Coordinates and Polar Curves)](https://reader031.vdocuments.site/reader031/viewer/2022012401/544ce01eb1af9f8d338b466c/html5/thumbnails/229.jpg)
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