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Math Help - Find asymptotes and sketch in polar

  1. #1
    DBA
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    Find asymptotes and sketch in polar

    Find the asymptotes by expressing the rectangular coordinates x and y as function of theta, then sketch the curve in polar.

    <br />
r= cot (\theta )<br />

    <br />
y= r*sin (\theta )<br />
    <br />
x= r*cos (\theta )<br />


    <br />
r= cot(\theta)<br />
    <br />
r= \frac{cos(\theta)}{sin(\theta)} |multipy both sides by sin(\theta)
    <br />
r sin(\theta) = \frac{cos(\theta)}{sin(\theta)} sin(\theta)<br />
    <br />
y = cos(\theta)<br />

    How do I get the horizontal asymptote from here?


    Then
    <br />
r= cot(\theta)<br />
    <br />
r= \frac{cos(\theta)}{sin(\theta)} |multipy both sides by cos(\theta)
    <br />
r cos(\theta) = \frac{cos(\theta)}{sin(\theta)} cos(\theta)<br />
    <br />
x= \frac{cos^{2}(\theta)}{sin(\theta)}<br />

    How do I get the vertical asymptote from here?
    Thanks!
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    To change the equation into rectangular form remember that r=\sqrt{x^2+y^2}

    Multiply both sides of the equation by r

    r=\frac{\cos(\theta)}{\sin(\theta)} \iff r\cdot r\sin(\theta)=r\cos(\theta) \iff \sqrt{x^2+y^2}\cdot y=x

    square one more time to get

    y^2(x^2+y^2)=x^2 \iff y^2=\frac{x^2}{x^2+y^2}

    To find the horizontal asymptotes take the limit as x \to \pm \infty

    For the vertical take the implicit derivative; solve for y' then set the denominator equal to 0.
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  3. #3
    DBA
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    Thanks for your answer, but I still do not knw how to get the asymptotes.

    How do I do that with y^2?

    And why do I have to take the implicit derivative for the vertical asymptote?

    Thanks
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