How might I approximate this curve?

Question

Hopefully this isn't too vague, but I would like to know an intuitive approach to approximating a function that outlines the negative space of this function (The shapes that resemble parabolas) Something that outlines only the positives value would work, or the negative values. I assume it'll have to be some periodic function, but past that, I can't really think of an intuitive way of determining the curve.

**I edited the picture to make the scale visible (whoops)

Answer 1

It looks a lot like $y=\pm \sec x$ or $y=\pm \csc x$, which are the same but offset by $\frac \pi 2$. The values seem to run from something which might be $\pm 1$ to $\pm \infty$ without ever crossing into $(-1,1)$. The period is something that might be $\pi$ but without a scale we can't tell.

Answer 2

This is probaly not an answer.

I do not see why you would like to approximate since there is a closed form

$$S_p=\sum_{n=0}^p \cos(n\pi x)=\frac{1}{2} \csc \left(\frac{\pi }{2}x\right) \sin \left(\frac{2 p+1}{2} \pi x\right)+\frac{1}{2}$$

Have a look here and here.

Answer 3

$$f(x)=\frac{1}{\sin ^2\left(\frac{\pi x}{2}\right)}$$