Step 1 :Define the function \(f = \sin(z) \sin(\pi x) \cos(y)\)
Step 2 :Integrate \(f\) with respect to \(y\) from 0 to 1, resulting in \(\sin(1) \sin(z) \sin(\pi x)\)
Step 3 :Integrate the result with respect to \(x\) from 0 to \(\pi\), resulting in \(-\sin(1) \sin(z) \cos(\pi^2)/\pi + \sin(1) \sin(z)/\pi\)
Step 4 :Finally, integrate the result with respect to \(z\) from 0 to \(\pi/2\), resulting in \(-\sin(1) \cos(\pi^2)/\pi + \sin(1)/\pi\)
Step 5 :The final answer to the problem is \(\boxed{-\frac{\sin(1)\cos(\pi^2)}{\pi} + \frac{\sin(1)}{\pi}}\)