Proving Self-Intersection at (0,0) for x = t cos(t), y = (pi/2 - t) sin(t)

[rcmango]

Homework Statement

Prove that the curve x = t cos(t), y = (pi/2 - t) sin(t) has a self-intersection at the point (0,0)

Homework Equations

The Attempt at a Solution

not sure where to start with this one. Please help.

 

[jhicks]

Find two t's such that x(t1)=y(t1)=x(t2)=y(t2)=0. Both x(t) and y(t) take periodic visits to 0, so it shouldn't be too hard to find. The construction of x(t) and y(t) give clues as to which values are good to look at first.

 

[rcmango]

Okay, so if I'm looking for two t's that both are 0 at x and y, i should be looking at the sin and cos graph where both are 0, correct?

so one good point would be (x1, y1) are 0,0 itself.
and the other point is about 1.571, where both graphs touch this point at the x axis.

are these points okay?

okay what's next!

 

[Dick]

So what values of t are you talking about?

 

[rcmango]

0 and 1.571

unless there is something easier you can suggest other than 1.571

 

[Dick]

That will do it. I kinda like pi/2 better than 1.571 though. And since cos(t) and sin(t) are never zero at the same value of t, they are the only two.