Trigonometry - Trigonometric Modelling

The diagram below shows a planning map of a planting area that contains a walking path, \(DA\), \(AB\) and \(BC\), where \(DA = 34\) m, \(AB = 12\) m, \(\angle DAB = \angle BCD = 90^\circ\) and the acute angle \(\angle ADC = \theta\) can vary.

1. Show that \(L\) m, the length of the walking path can be expressed as \(L = 46 + 34 \sin \theta - 12 \cos \theta\).
2. Express \(L\) in the form \(p + R \sin(\theta - \alpha)\), where \(p\), \(R > 0\) and \(0^\circ < \alpha < 90^\circ\).
3. Given that the exact length of the walking path is \(62\) m, find the value of \(\theta\).