Conformal maps from a sector disk to a quadrant

Let \[ \Omega_1=\{z:|z|<1,\ 0<\arg z<\alpha<2\pi\}, \qquad \Omega_2=\{x+iy:x,y>0\}. \] (a) Find an explicit conformal map \(\Omega_1\to\Omega_2\).

2. How many conformal maps are there between these two domains? Prove your answer.