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Math Help - A couple things: Polar coordinate problems

  1. #1
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    A couple things: Polar coordinate problems

    Hello everyone:

    I'm reviewing for a test, and I got some questions regarding polar coordinates:

    Question one:

    Show that the point  (2, 3\pi/4) lies on the curve  r = 2 sin2\theta

    Specifically, when you simply "plug in" the  r and  \theta , it gives you a false equality. I know that there are multiple ways to express the point  (2, 3\pi/4) , and that apparently you must replace this point with an equivalent point in order to make the equality work. What exactly must I do to make this work?

    Question two:

    Find the intersections of this pair of curves:
     r = 1 + cos\theta and  r = 1 - cos\theta


    Thanks!
    Last edited by Skinner; November 2nd 2008 at 10:53 PM.
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  2. #2
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    Quote Originally Posted by Skinner View Post
    Hello everyone:

    I'm reviewing for a test, and I got some questions regarding polar coordinates:

    Question one:

    Show that the point  (2, 3\pi/4) lies on the curve  r = 2 sin2\theta

    Specifically, when you simply "plug in" the  r and  \theta , it gives you a false equality. I know that there are multiple ways to express the point  (2, 3\pi/4) , and that apparently you must replace this point with an equivalent point in order to make the equality work. What exactly must I do to make this work?

    Question two:

    Find the intersections of this pair of curves:
     r = 1 + cos\theta and  r = 1 - cos\theta


    Thanks!
    Q1 The point specified by [2, 3pi/4] is also specified by [2, -5pi/4].


    Q2 You should first draw both curves.

    Algebra tells you that the curves intersect at [1, pi/2] and [1, 3pi/2], that is, at (0, 1) and (0, -1):

    1 + \cos \theta = 1 - \cos \theta \Rightarrow \cos \theta = 0 \Rightarrow \theta = \frac{\pi}{2}, \, \frac{3 \pi}{2}.

    But drawing both curves will tell you that the curves also intersect at the origin. The reason algebra doesn't tell you this is that this intersection point has polar coordinates [0, pi] for the first curve and [0, 0] for the second curve .... that is, this point has no simultaneous solution for \theta.
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