Trigonometry - Trigonometric Modelling

The diagram below shows a robotic arm \(PQR\) such that \(PQ\) makes an angle of \(\theta\) with the \(y\)-axis, where \(0^\circ < \theta < 90^\circ\) and the angle \(PQR = 90^\circ\). The lengths of \(PQ\) and \(QR\) are \(6\) m and \(4\) m respectively.

1. Given that the vertical length of the robotic arm is \(L\) m,
   1. show that \(L = 4 \sin \theta + 6 \cos \theta\), and
   2. express \(L\) in the form \(R \sin (\theta + \alpha)\), where \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).
2. Based on your answers in (i),
   1. find the maximum length of \(L\) and find the corresponding value of \(\theta\), and
   2. find the value(s) of \(\theta\) when \(L = 7\) m.