Parametric function - double points

[JPC]

Homework Statement

The parametric function :
x = cos(5t)
y= cos(3t)
t belongs to R

Question : find the coordinates (x, y) of the double points

Homework Equations

The Attempt at a Solution

OK so first of all,i find an interval of t where to study
- periodic of 2Pi
- M(t) = M(-t)
- M(t+Pi) is the symmetric of M(t) by point O
- M(Pi-t) is the symmetric of M(t) by point O

So i decide to study on E=[0, Pi/2], and then just do the symmetric by O to get it on [0, Pi]
and since M(t) = M(-t), then i have it on [-Pi, Pi], which is an interval of 2Pi, meaning that i have it on R.

here is a graph

Now i need to get the coordinates (x, y), of the 4 "double points"
knowing that there is a symmetry by O, i just need to find the coordinates of 2 non-symmetric of the 4.

but now i am stuck, because we have just spent one class all this, what is the method to find these double points ?

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Logically, if 'u' belongs to E=[0, Pi/2] and 'w' belongs to F=[Pi/2, Pi],
we are looking for x(u) = x(w) AND y(u) = y(w)

but i was thinking we might need to use the symmetry too
so, we consider 'z' belongs to E=[0, Pi/2], with
x(w) = - x(z)
y(w) = - y (z)

I have tried with this, but doesn't give me any concluding results, since it just expresses me w with z :D

 

[HallsofIvy]

Yes, cosine is periodic with period 2\pi. Looking at a graph of y= cos(x), you should also see that it is true that cos(x)= cos(2\pi- x). Thus, to have x= cos(5t)= cos(5s) and y= cos(3t)= cos(3s) we must have either 5t= 5s+ 2n\pi and 3t= 3s+ 2k\pi or 5t= \2npi- 5s and 3t= 2k\pi- 5s.

In the first case we have 5(t-s)= 2n\pi and 3(t-s) 2k\pi which means t-s= M(2\pi) is divisible by both 3 and 5: divisible by 15.

 

[JPC]

Thank you
but from there how do i find the two values of 't' in E=[0, Pi/2] ?
by looking at the graph, with x=0.5 and x=-0.5, its easy to say the two values of t for E=[0, Pi/2] are : Pi/15 and 2Pi/15
(once i have the (x, y) coords of these 2 double points, its easy to get those of the 2 others by (-x, -y) )