Place where parametric curve itself itself

  • #1

Homework Statement


Find the place where the parametric curve intersect itself

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



Homework Equations





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]
and
[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
 

Answers and Replies

  • #2
HallsofIvy
Science Advisor
Homework Helper
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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)?
 
  • #3
Ah, now I know what you meant, thanks!
 

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