Find a conformal map of the half-strip $\displaystyle \{ -\frac{\pi}{2}<\text{Re}(z)<\frac{\pi}{2}, \text{Im}(z)>0 \}$ onto the open unit disk that maps $\displaystyle \frac{\pi}{2}$ to $\displaystyle -i$, and 0 to +1. Where does $\displaystyle \infty$ go under this map?

The back of the book says that $\displaystyle w=-\frac{\sin(z)-i}{\sin(z)+i}, w(\infty)=-1$ works. However, I don't see how they got this map. That is what is I need help with. Thanks.