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Place where parametric curve itself itself

  1. Mar 29, 2008 #1
    1. The problem statement, all variables and given/known data
    Find the place where the parametric curve intersect itself

    [tex] x = 1-2cos^{2}t [/tex]
    [tex] y = tant(1-2cos^{2}t)[/tex]

    2. Relevant equations

    3. The attempt at a solution
    So I started with the x values..

    [tex] 1-2cos^{2}t_{1} = 1-2cos^{2}t_{2} [/tex]

    By canceling the same stuff on both sides, I got
    [tex] cos^{2}t_{1} = cos^{2}t_{2} [/tex]

    Then I tried with y
    I rewrote y in a different form.

    [tex] y_{1} = tan t (x_{1}) [/tex]
    [tex]y_{2} = tan t(x_{2}) [/tex]

    This is possible since y already contains an expression for x.

    Since the curve intersect itself, we know x1 must equal x2 so they cancel out.

    then I am left with [tex] tant_{1}= tant_{2} [/tex]

    but I can't solve for t

    Appreciate any help. Thanks
  2. jcsd
  3. Mar 29, 2008 #2


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    Science Advisor

    Why can't you? Since tangent is periodic with period [itex]\pi[/itex], but one-to-one within each period, tan(t1)= tan(t2) requires that [itex]t_1= t_2+ n\pi[/itex]. Now, which of those values satifies cos2(t1)= cos2(t2)?
  4. Mar 29, 2008 #3
    Ah, now I know what you meant, thanks!
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