9.3 Polar Coordinates
1. ,
http://matrix.skku.ac.kr/cal-lab/cal-9-3-1.html
def Polar(r,theta): #converts Polar to Cartesian Coordinates CartC = ([r*cos(theta),r*sin(theta)]); return CartC; |
pt=Polar(-1,pi); vector(pt) |
(1, 0)
list_plot([pt], aspect_ratio=1,xmin=-1, xmax=2, ymin=-1, ymax=1) |
2. .
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4. .
5-8. The Cartesian coordinates are given. Find two other pairs of polar coordinates of the point, one with and other .
5. .
, and .
Since is in the fourth quadrant, the polar coordinates are
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6. .
, and .
Since is in the second quadrant, the polar coordinates are
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7. .
, and .
Since is in the fourth quadrant, the polar coordinates are
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8. .
9. Find a formula for the distance between the points with polar coordinates and .
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10-11. Find a polar equation for the curve represented by the given Cartesian equation.
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11. .
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12. .
13-14. Find a Cartesian equation for the curve represented by the given polar equation.
13. .
, a circle of radius centered at . The first two equations are actually equivalent since
. But gives the point when .
Thus, the single equation is equivalent to the compound condition ().
14. .
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16-25. Sketch the curve with the given polar equation.
16. .
17. .
http://matrix.skku.ac.kr/cal-lab/cal-9-3-17.html
theta=var('theta'); polar_plot(sin(theta), (0, 2*pi)).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2) |
18. .
http://matrix.skku.ac.kr/cal-lab/cal-9-3-18.html
r=var('r'); polar_plot(3,(0, 2*pi)).show(aspect_ratio=1, xmin=-5, xmax=5, ymin=-5, ymax=5) |
19. .
http://matrix.skku.ac.kr/cal-lab/cal-9-3-19.html
theta=var('theta'); polar_plot(theta, (0, 2*pi)).show(aspect_ratio=1, xmin=-10, xmax=10, ymin=-10, ymax=10) |
20. .
http://matrix.skku.ac.kr/cal-lab/cal-9-3-20.html
theta=var('theta'); polar_plot(1-2*cos(theta), (0, 2*pi)).show(aspect_ratio=1, xmin=-4, xmax=4, ymin=-4, ymax=4) |
21. .
http://matrix.skku.ac.kr/cal-lab/cal-9-3-21.html
theta=var('theta'); polar_plot(sin(theta/2), (0, 2*pi)).show(aspect_ratio=1, xmin=-2, xmax=2, ymin=-2, ymax=2) |
22. .
http://matrix.skku.ac.kr/cal-lab/cal-9-3-22.html
theta=var('theta'); polar_plot(sin(4*theta), (0, 2*pi)).show(aspect_ratio=1, xmin=-1, xmax=1, ymin=-1, ymax=1) |
23. .
24. .
25. .
26. Show that the polar curve (called a conchoid) has the line as a vertical asymptote by showing that . Use this fact to help sketch the conchoid.
http://matrix.skku.ac.kr/cal-lab/cal-9-3-26.html
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Now, , so .
Also, , so
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Therefore, is a vertical asymptote.
theta=var('theta'); polar_plot(4+2*sec(theta), (0, 2*pi)).show(aspect_ratio=1, xmin=-5, xmax=15, ymin=-10, ymax=10) |
27. Show that the curve (also a conchoid) has the line as a horizontal asymptote by showing that Use this fact to help sketch the conchoid.
http://matrix.skku.ac.kr/cal-lab/cal-9-3-27.html
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Now, , so .
Also, , so .
Therefore, is a horizontal asymptote.
theta=var('theta'); polar_plot(2-csc(theta), (0, 2*pi)).show(aspect_ratio=1, xmin=-15, xmax=15, ymin=-10, ymax=10) |
28.Show that the curve (called a cissoid of Diocles) has the line as a vertical asymptote. Show also that the curve lies entirely within the vertical strip . Use these facts to help sketch the cissoid.
29-32. Find the slope of the tangent line at the given point.
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32. .
33-36. Find the points on the given curve where the tangent line is horizontal or vertical.
33. .
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So the tangent is vertical at and .
. So the tangent is horizontal at and .
34. .
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36. .
37.Show that the polar equation represents a circle and find its center and radius.
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Then the polar equation represents a circle and its center : , radius :.