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Math Help - Polar graph: stuck on sketching a cardioid

  1. #1
    Member ssadi's Avatar
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    Polar graph: stuck on sketching a cardioid, something about chords

    Sketch the cardioid with polar equation r=2(1+cos theta)
    Prove that for all the chords POQ of the cardioid which pass through the pole O, the length of PQ is constant.
    How do I do the green part?
    I have gone as far as to deduce that if P (r1, theta1) and Q is (r2, theta2)
    Then theta2=theta1-180
    r2=2+2cos (theta1-180)
    There isn't given a formula in the book to find the distance between two polar coordinates, I found one on net:

    Here is the sketch of the graph:

    I am stuck here for over two hours. Emergency help...I am drow...(Oops I gulped up some water)
    Last edited by ssadi; October 17th 2008 at 01:53 PM. Reason: A more elaborate title
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  2. #2
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    Quote Originally Posted by ssadi View Post
    There isn't given a formula in the book to find the distance between two polar coordinates, I found one on net:
    You don't need any formula from the internet... The distance from O to P(r,\theta) is r. So what you have to do is add the values of r for \theta_1 and \theta_2=180-\theta_1 (in degrees), and check that it does not depend on \theta_1.
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  3. #3
    Member ssadi's Avatar
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    Quote Originally Posted by Laurent View Post
    You don't need any formula from the internet... The distance from O to P(r,\theta) is r. So what you have to do is add the values of r for \theta_1 and \theta_2=180-\theta_1 (in degrees), and check that it does not depend on \theta_1.
    theta2=theta1-180 according to my calculations. I am giving it a try.
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  4. #4
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    Quote Originally Posted by ssadi View Post
    theta2=theta1-180 according to my calculations. I am giving it a try.
    Sorry I made a mistake, in fact \theta_2=\theta_1+180. Because \theta_2-\theta_1 is a flat angle.
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  5. #5
    Member ssadi's Avatar
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    Quote Originally Posted by Laurent View Post
    You don't need any formula from the internet... The distance from O to P(r,\theta) is r. So what you have to do is add the values of r for \theta_1 and \theta_2=180-\theta_1 (in degrees), and check that it does not depend on \theta_1.
    It worked. Thanks.
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  6. #6
    Member ssadi's Avatar
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    Quote Originally Posted by Laurent View Post
    Sorry I made a mistake, in fact \theta_2=\theta_1+180. Because \theta_2-\theta_1 is a flat angle.
    Does that mean I am wrong here? I have verified it with calculator, my formula worked for polar -pi<theta<pi
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  7. #7
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    Quote Originally Posted by ssadi View Post
    Does that mean I am wrong here? I have verified it with calculator, my formula worked for polar -pi<theta<pi
    No, indeed, you're right as well... I considered the graph defined for 0<\theta_2<2\pi, while you considered -\pi<\theta_2<\pi, that's why. Forget my previous post, I was correcting my first one, not yours in fact.
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  8. #8
    Member ssadi's Avatar
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    Quote Originally Posted by Laurent View Post
    No, indeed, you're right as well... I considered the graph defined for 0<\theta_2<2\pi, while you considered -\pi<\theta_2<\pi, that's why. Forget my previous post, I was correcting my first one.
    I guessed so, that your limits might be different. But my problem is solved anyway.
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