I have the following map, $f: \mathbb{R} \rightarrow \mathbb{R^2}$ given by $f(\theta) = (cos\theta, sin\theta)$. The differential of the map is $$df_\theta(\partial_\theta) = -\sin \theta \partial_1 + \cos \theta \partial_2$$, where $\partial_{1,2}$ denote differentitial w.r.t $x_1,x_2$ on $\mathbb{R^2}$.
The answer says that the differential is non-zero for all $\theta$ and so it's injective for all $\theta$. I don't understand why it is injective since it is a periodic function. Probably I am missing something trivial but I don't see what.