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Math Help - Polar Coordinates Problem

  1. #1
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    Polar Coordinates Problem

    Hey, I have a polar coordinate problem here that feels like it should be much easier than I am making it out to be.

    Find all points of intersection of the following curves, given in Polar Coordinates. r^2 = cos(2\theta) \ and \ r = 1 - sin(\theta)
    So I started by setting them equal to each other to find the angles \theta at which they intersect.

     cos(2\theta) = (1-sin\theta)^2

    1-2sin^2\theta = 1-2sin\theta + sin^2\theta

    sin\theta = \frac{2}{3}

     \theta \approx 0.729 rad

    The problem is that by graphing it appears that there is 5 points of intersection, and I only have 1 floating point approximation of theta. What did I miss? Thanks for the help!
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  2. #2
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    Your algebra missed a beat. What did you do with \sin(\theta) = 0?

    By symmetry, you can get another one from the one you have. Think about Reference Angles.

    By more symmetry, you can get two more from the one you missed. That makes five (5). Your graphing is exemplary.

    Note: Check your Domain. You may not get all five (5).
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  3. #3
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    Quote Originally Posted by TKHunny View Post
    Your algebra missed a beat. What did you do with \sin(\theta) = 0?

    By symmetry, you can get another one from the one you have. Think about Reference Angles.

    By more symmetry, you can get two more from the one you missed. That makes five (5). Your graphing is exemplary.

    Note: Check your Domain. You may not get all five (5).
    Wooow, what a miss that was... Gotta watch for that in the future.

    So with sin(\theta)=0 \ \ \rightarrow \ \ \theta = 0, \pi, 2\pi, we are at 4. Now by graphing y=\frac{2}{3} \ and \ y = sin(\theta) I see that the other intersection point lies at \theta \approx 2.41 rad. But I have no idea how to use symmetry to get that, as it doesn't seem to be an addition of any proper fraction of pi. Any hints on how to grab that last intersection algebraically?

    Thank you for your help!
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  4. #4
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    \sin(\theta) = \sin(\pi - \theta)

    Reference Angles!!
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  5. #5
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    Note to self - Review high school trig.

    Thanks very much
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