Trigonometry - Trigonometric Modelling

The diagram shows a robotic arm \(PQR\) such that \(PQ\) makes an angle of \(\theta\) with the \(y\)-axis, where \(0{}^\circ < \theta < 90{}^\circ\). \(QR\) makes an angle of \(\theta\) with the horizontal. 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 \(L=4\sin \theta + 6\cos \theta\).
   2. express \(L\) in the form \(R\sin(\theta + \alpha){}^\circ\), where \(R>0\) and \(0{}^\circ < \theta < 90{}^\circ\).
2. Hence,
   1. state the maximum length of \(L\) and find the corresponding value of \(\theta\).
   2. find the values of \(\theta\) when \(L=7\) m.