Prove that the map $F:\mathbb{R}\to S^1$, $F(t)=(\cos t,\sin t)$ is $C^\infty$.

Question

Why isn't smoothness of $(\cos t,\sin t)$ as a map from $\mathbb{R}$ to $\mathbb{R}^2$ enough? IS it because when we have $S^1$, the codomain is changed? But that shouldn't affect smoothness, right? Why do we need to bring in charts to prove smoothness?