Trigonometry

A rectangle \(ABCD\) is inscribed in a semicircle of radius \(5\text{ cm}\). Its lower side \(AB\) lies on the diameter and is centred at \(O\). Its upper vertices \(C,D\) lie on the arc.
The radius \(OC\) makes an angle \(\theta\) with the horizontal, where \(0^\circ<\theta<90^\circ\).

1. Show that the rectangle’s perimeter is \( P=20\cos\theta+10\sin\theta. \)

2. Express \(P\) as \(R\cos(\theta-\alpha)\), and find its exact maximum.

3. Find the permitted value of \(\theta\) when \(P=20\text{ cm}\).