Consider he action of $U(1)$ on $S^3$ by $(z_1,z_2)\cdot e^{i\phi}\mapsto(z_1e^{i\phi},z_2e^{i\phi})$. This action is clearly free and proper. Therefore we can consider $S^3$ as a principal bundle with structure group $U(1)$ over $S^3/U(1)\cong S^3/S^1\cong\mathbb{C}P^1$.
I know that we have charts for $\mathbb{C}P^1$ given by $U_i=\{[z_1,z_2]|z_i\neq0\}$. How do I compute the transition function $\varphi_{U_1U_2}:U_1\cap U_2\rightarrow U(1)$?