dg.differential geometry - Transition Functions of the Principal Bundle $SU(2) to mathbb{CP}^1$

I've been trying to understand principal bundles, and to that end have been looking at the bundle
$$
pi: SU(2) to mathbb{CP}^1,~~~ (a_{ij}) mapsto [a_{11},a_{21}],
$$
with fibre $U(1)$. I assumed that the bundle would be trivial over the standard nbds $U_1,U_2 subset mathbb{C}$, but can't seem to identify the local trivializations. Now
$$
pi^{-1}(U_1) = {left( array{a & - overline{b}\
b & overline{a}} right)|~ a neq 0}, ~~~ pi^{-1}(U_2) = {left( array{a & - overline{b}\
b & overline{a}} right)|~ b neq 0},
$$
and any trivialization $alpha_1:pi^{-1}(U_i) to U_i times U(1)$, will map
$$
alpha_1:left( array{a & - overline{b}\
b & overline{a}} right) mapsto ([a,b],h_{a,b}^1),
$$
for some $h_{a,b}^1$. Defining $h^1_{a,b} = arg(a) = frac{a}{|a|}$, and similarly $h^2$, works, but then the transition functions are not in $U(1)$.