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. .

.


3. .

.


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

      ⑴ : , ⑵ : .


6. .

, and .

      Since is in the second quadrant, the polar coordinates are

      ⑴ : , ⑵ : .


7. .

, and .

      Since is in the fourth quadrant, the polar coordinates are

      ⑴ : , ⑵ : .


8. .


9. Find a formula for the distance between the points with polar coordinates and .

.


10-11. Find a polar equation for the curve represented by the given Cartesian equation.


10. .   

.


11. .

.


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. .

.


15. .

.


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 

.

      Now, , so .

      Also, , so

      .

      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 

.

    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.


29. .

 .


30. .

.


31. .

.


32. .


33-36. Find the points on the given curve where the tangent line is horizontal or vertical.


33. .  

.

     .

     So the tangent is vertical at and .

     . So the tangent is horizontal at and .


34. .


35. .


36. .


37.Show that the polar equation represents a circle and find its center and radius.

.

      Then the polar equation represents a circle and its center : , radius :.