To prove that $f$ ( the complex conjugation map )is orientation reversing when considered as a map from $S^1\to S^1$

Question

Considering $S^1$ as a subset of $\mathbb{C}$ I need to show that the complex conjugation map is orientation reversing. Thinking $\mathbb{C}$ as $\mathbb{R^2}$ the conjugation map gives $f(x,y)=(x,-y)$ so it will be orientation reversing by calculating the Jacobian but how to claim it for $S^1$ that I am unable to do. Any hints or help is very much needed.