Show that $f(z) = 5\sin(z) - \exp(z)$ has exactly one zero in a region.

Question

Let $f(z) = 5\sin(z) - \exp(z)$.

To show that this function has exactly one zero in the square $\bigl\{z \in \mathbb{C} : \lvert\mathcal{R}(z)\rvert < \pi/2 ,\:\lvert\mathcal{I}(z)\rvert < \pi/2 \bigr\}$, I was thinking of using Rouché's theorem. However, I can't seem to find functions with a zero in the square region that fit the theorem. How does one approach this problem?