How do I find the 2 slopes at which this parametric function crosses itself?

Question

I have a parametric function. If you graph it, you'll find that it looks like a figure 8.

x(t) = 2sin(2t)
y(t) = 8sin(t)

How do I find the slopes of the function when it intersects with itself again (graph this to see what I mean)? Should I convert it to a non-parametric function first?

Answer 1

There’s no need to abandon parametric form:

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}=\frac{8\cos t}{4\cos 2t}=\frac{2\cos t}{\cos 2t}\;.$$

Now just find (in terms of $t$) where it crosses itself.

Answer 2

Let $x_i=2\sin 2t_i$ and $y_i=8\sin t_i,$

to find the points where the curve crosses it self, we need $x_i=x_j$ and $y_i=y_j$ for some $i\ne j$

So, $\sin t_i=\sin t_j--->(1)$ and $\sin 2t_i=\sin 2t_j--->(2)$

$(1)$ gives, $t_i=n\pi+(-1)^nt_j$

$(2)$ gives, $2t_i=m\pi+(-1)^m2t_j$ where $m,n$ are integers.

If $n=2q, t_i=2q\pi+t_j-->(3)$

If $n=2q+1, t_i+t_j=(2q+1)\pi--->(4)$

If $m=2r, 2t_i=2r\pi+2t_j\implies t_i=r\pi+t_j--->(5)$

If $m=2r+1, 2t_i=(2r+1)\pi-2t_j\implies t_i+t_j=(2r+1)\frac{\pi}2--->(6)$

We can only combine, $(3)$ and $(5),$ so that $t_i=2s\pi+t_j$