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Parametric equations and tangent(s)

  1. Oct 14, 2007 #1
    1. The problem statement, all variables and given/known data

    At what point does the curve [tex]x = (1-2cos^2(t)[/tex], [tex]y=(tan(t))(1-2cos^2(t))[/tex] cross itself? Find equations of both tangents at that point.



    2. Relevant equations



    3. The attempt at a solution

    To begin, I figured that x=y when it crosses itself, so I set x=y and got [tex]1 = tan(t)[/tex], so [tex]t=\pi/4[/tex]

    Next, I decided to solve for dy/dx:

    [tex]dx/dt=4cos(t)sin(t)[/tex]
    [tex]dy/dt=4sin^2(t)+sec^2(t)-2[/tex]

    [tex]dy/dx=\frac{(dy/dt)}{(dx/dt)}[/tex]

    At that point, I plugged [tex]\pi/4[/tex] into my eqn for dy/dx and got that the slope at that point is [tex]1[/tex]

    Additionally, I solved for x and y at [tex]\pi/4[/tex] and got that both x and y are 0 when [tex]t=\pi/4[/tex]. Using these points and the slope, I got that the equation of the tangent line is:

    [tex]y=x[/tex]


    BUT

    That's just one of them. If the curve is crossing itself once, it clearly has two tangent lines at that point, but I'm not sure how to find it. Is the slope of that line just [tex]-1[/tex]? I guess it might be, but it doesn't seem right to just assume that it is.

    Should I have said that [tex]t=n\pi + \pi/4[/tex]? If that is the case, would I just say that [tex]t=5\pi/4[/tex] AND [tex]t=\pi/4[/tex]? Of course, that assumes that I did the rest of the problem properly, and I'm not even sure if I did. I was very ill all last week, so I have no lecture/recitation notes to go by, and my book doesn't have very many examples for me to practice from.

    I'm sorry if this is a bad question. :( Thanks for looking!
     
    Last edited: Oct 14, 2007
  2. jcsd
  3. Oct 14, 2007 #2
    im not 100% on this, but i dont think its when x=y
    i think it means there are two different values of t that give the same x and y values?
    but i could be wrong
     
  4. Oct 14, 2007 #3

    EnumaElish

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    Given x = f(t), y = g(t), I think the question is to find t0 and t1 such that f(t0) = f(t1) and g(t0) = g(t1). Or to find t and s so that f(t) = f(t+s) and g(t) = g(t+s).
     
  5. Oct 14, 2007 #4

    HallsofIvy

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    Exactly. Not "y= x" but f(t)= f(s) , g(t)= g(s). (Yes, you could find s such that the second point is t+ s, but I think that's unnecessary and too complicated. And I expect EnumElish eant f(t)= f(s, g(t)= g(s) anyway.)

    Shouldn't be too difficult to find all solutions to f(t)= 1- 2cos2(t)= 1- 2cos2(s)= f(s). Now which of those also satisfy g(t)= tan(t)(1-2cos2(t))= tan(s)(1- 2cos2(s))?
     
  6. Oct 14, 2007 #5
    delete this post plz
     
    Last edited: Oct 14, 2007
  7. Oct 15, 2007 #6
    Ah, I think I see now. Thanks for the help!

    I guess it wasn't enough to simply say:

    [tex](1-2cos^2(t) = (tan(t))(1-2cos^2(t))[/tex] and solve for t like I was doing. However, the answer I got for that came out close to what I got for [tex]1- 2cos2(t)= 1- 2cos2(s)[/tex]

    I appreciate the help. :)
     
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