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

• rcmango
In summary, to prove that the curve x = t cos(t), y = (pi/2 - t) sin(t) has a self-intersection at the point (0,0), we need to find two values of t where both x and y are equal to zero. The values of t that satisfy this are 0 and pi/2. These are the only two values that will prove the self-intersection at the point (0,0).
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)

## The Attempt at a Solution

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.

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!

So what values of t are you talking about?

0 and 1.571

unless there is something easier you can suggest other than 1.571

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.

Last edited:

## 1. How do you prove self-intersection at (0,0) for the given functions?

To prove self-intersection at (0,0), we can use the concept of limit. We will take the limit of the functions as t approaches 0 from both the positive and negative sides. If the limit on both sides is equal to (0,0), then we can conclude that the functions intersect at (0,0).

## 2. Can you explain the significance of (0,0) in the given functions?

The point (0,0) is significant because it is the origin, where the x and y axes intersect. It is also the starting point for the parametric equations given, and therefore, determining if there is self-intersection at this point is crucial.

## 3. What does it mean for the functions to have self-intersection?

Self-intersection means that the functions intersect with themselves at a particular point. In other words, there are two different values of t that result in the same (x,y) coordinates, indicating that the functions overlap or cross each other.

## 4. How does the given function's graph look like when there is self-intersection at (0,0)?

When there is self-intersection, the graph of the functions will have an overlapping or crossing point at (0,0). This will result in a loop or a cusp shape in the graph, indicating that the functions are intersecting with themselves at the origin.

## 5. Is it possible for the given functions to have self-intersection at a point other than (0,0)?

Yes, it is possible for the functions to have self-intersection at a point other than (0,0). This can happen if the limit of the functions at that particular point is equal to that point. However, for the given functions, we are specifically trying to prove self-intersection at (0,0).

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