10.5.40 Conformal mappings

Problem: Find the image of domain \[ D=\left\{|\operatorname{Re} z|<\frac{\pi}{2}\right\} \cup\{\operatorname{Im} z>0\} \backslash\{z=i t, t \in(0 ; 1]\} \] for the mapping \( f(z)=\sin z \).