 Reduction to the standard modular region

Given $\tau \in \mathfrak{h}$, choose a primitive pair $(c,d)$ minimizing $|c\tau +d|$ and apply a modular matrix with bottom row $(c,d)$. This maximizes the imaginary part within the orbit. Translation puts the real part in $[-1/2,1/2]$, and inversion then shows that the imaginary part is at least $\sqrt{3}/2$.