Trigonometry - Trigonometric Modelling

The diagram below shows the chords \(PR\) and \(QS\) that intersect at the point \(M\). Given that \(PQ = 7\) cm, \(RS = 24\) cm, the angle \(\angle RMQ = 90^\circ\) and the angle \(\angle PRS = \theta\), where \(\theta\) varies.

1. Prove that \(PR = 24 \cos \theta + 7 \sin \theta\).
2. Express \(PR\) in the form of \(R \cos (\theta - \alpha)\), where \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).
3. Find the maximum length of \(PR\) and the corresponding value of \(\theta\).