I've tried a few methods but I can't seem to work this one out.
Consider the charts $$f(s) = (\cos s, \sin s) \in \mathbb{R}^2$$ for $-\pi < s < \pi$ and $$g(t)=(\frac{2t}{t^2 + 1}, \frac{t^2 - 1}{t^2 + 1})$$ for $t \in \mathbb{R}$. Are these charts on the circle compatibly oriented?