conformal map, half-strip

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

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