2026 HGV P2 Q6

The diagram shows the vertical cross section \(ABCD\) of a greenhouse, where \(AB\) is on horizontal ground and \(AD\) is a doorway of height \(H\) m. The lengths of \(BC\) and \(CD\) are 20 m and 14 m respectively. \(CB\) is inclined at an angle \(\theta^\circ\) to \(AB\), where \(0^\circ\le\theta\le90^\circ\). Angles \(BCD\) and \(BAD\) are right angles.

1. Show that \(H=20\sin\theta-14\cos\theta\).[2]
2. Express \(H\) in the form \(R\sin(\theta-\alpha)\), where \(R>0\) and \(0^\circ\le\alpha\le90^\circ\).[3]
3. Find the value of \(\theta\) for which the height of the doorway is 4 m.[3]