What function(s) satisifies $f(\theta)=-f(\theta+2 \pi )$?

Question

This may be a trivial question, but perhaps someone can give a detailed answer. I'm looking for a periodic function that satisfies $$f(\theta)=-f(\theta+2\pi)$$ where $\theta$ is an angle in radians.

I have no idea if such a function exists, or how to create one. Is there any function (or class of functions) that fits this description?

Answer 1

You could take $$f(\theta)=\sin\Bigl(\frac\theta2\Bigr)\ ,$$ among others.

Alternatively, let $f(\theta)$ be anything you like for $0\le\theta<2\pi$, then define $$f(\theta)=-f(\theta-2\pi)$$ for $2\pi<\theta<4\pi$, then $$f(\theta)=f(\theta+4\pi)$$ for all $\theta$.