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Math Help - polar to rectangular equation

  1. #1
    Junior Member
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    Sep 2009
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    polar to rectangular equation

    How do I convert the following equation from polar to rectangular form?

    r^2 = sin\Theta

    I know r^2 = x^2 + y^2 , but I'm not sure how to eliminate sin\Theta. I know rsin\Theta= y, but multiplying both sides by r just muddies it further for me.
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  2. #2
    Senior Member
    Joined
    Feb 2010
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    y = r \sin \theta
    y = r\, r^2
    y^2 = r^2r^4
    y^2 = (x^2 + y^2)^3

    Let u := y^2. Let v:= x^2.

    u = u^3 + v^3 + 3u^2v + 3 uv^2
    u^3 +3vu^2 + (3v^2-1)u + v^3 = 0

    Ummm.... here's where I lose it...

    Let \alpha := \sqrt{3} \sqrt{27x^4 - 4} - 9x^2.

    |y| = \big( \sqrt[3]{\alpha/18}+\sqrt[3]{2/3\alpha} - x^2 \big)^{1/2}

    ... I think (I used the cubic formula). There might also be more solutions (pretty sure there are...)
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