Consider right triangle \(OPQ\) given in the following figure, and assume that the length of the hypotenuse is \(OP = r\) for some constant \(r \gt 1\text{.}\) Let point \(M\) lie on \(\overline{OP}\) (the line segment between \(O\) and \(P\)) in such a way that \(OM = 1\text{,}\) and let point \(N\) lie on \(\overline{OQ}\) so that \(\angle ONM\) is a right angle, as pictured. In addition, assume that point \(O\) corresponds to \((0,0)\text{,}\) point \(Q\) to \((x,0)\text{,}\) and point \(P\) to \((x,y)\) so that \(OQ = x\) and \(PQ = y\text{.}\) Finally, let \(\theta\) be the measure of \(\angle POQ\text{.}\)