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Parametric -> Implicit Equations

  1. Apr 11, 2014 #1
    Hi guys,

    I have done what I can with the following:

    Given a parametric curve x = xsint, y = sin(2t) where t is in R.

    Find an implicit equation of this curve.

    MY ANSWER:
    y = 2costsint = costx

    Therefore sint = x/2, cost = y/x

    sin^2(t) + cos^2(t) = x^2 / 4 + y^2 / x^2 = 1

    Would this be right?
    I understand that x != 0 but why can't I just multiply every value underlined by 4x^2?

    Thank you, any input is appreciated!
     
    Last edited: Apr 11, 2014
  2. jcsd
  3. Apr 11, 2014 #2

    LCKurtz

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    Please correct the typo's so we know what the actual equations are.
     
  4. Apr 11, 2014 #3

    LCKurtz

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    I will assume, from your work below, that the equations are ##x=2\sin t,~y=\sin(2t)##

    Yes, you can multiply through by ##4x^2##. The original equations certainly allow ##x=0## and the only problem is that you divided by ##x## in your solution. You could have derived the same equation as you get when you multiply through by ##4x^2## without ever dividing by ##x##. So you should give the answer as $$
    4x^2 = 4y^2+x^4$$or some variation of that.
     
  5. Apr 11, 2014 #4

    SammyS

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    Do you mean
    x = sin(t)

    and

    y = sin(2t)

    ?​
     
  6. Apr 11, 2014 #5
    Yes sorry That is what I meant.

    How do you derive it from the original equation?
    x=2sint, y=sin(2t) without dividing by x?
     
  7. Apr 11, 2014 #6

    LCKurtz

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    $$x^2+y^2=4\sin^2 t + (2\sin t\cos t)^2=4\sin^2t(1+\cos^2 t)
    =4\sin^2 t(2-\sin^2 t) = x^2(2-\frac{x^2} 4) = \frac{x^2(8-x^2)}{4}$$
     
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