The inverse function of \(F\) is equal to \(F^{-1}(x,y,z) = (y \cdot \sin(2z) + x \cdot \cos(2z), 2z)\).
Since \(F(u,v) = (u \cdot \cos(v), u \cdot \sin(v), 0.5 v)\), if we define \(F^{-1}(x,y,z) = (u,v)\) it implies that \(v=2z\). To determine the value of \(u\) we leverage the fact that \(\sin^2(v) + \cos^2(v) = 1\) which implies: \[\begin{aligned} u &= u \cdot (\sin^2(v) + \cos^2(v))\\ &= (u \cdot \sin(v)) \cdot \sin(v) + (u \cdot \cos(v)) \cdot \cos(v) \\ &= y \cdot \sin(v) + x \cdot \cos(v) \\ &= y \cdot \sin(2z) + x \cdot \cos(2z). \end{aligned}\]