Trigonometry - Basics of Trigonometric Ratios

A rectangle is inscribed in a circle of radius 3 cm and centre O,

The point \(P\ (x,y)\) is a vertex of the rectangle and also lies on the circle. The angle between (OP) and the x-axis is \(\theta\) radians, where \(0 \leq \theta \leq \frac{\pi}{2}\).

1. Write down an expression in terms of \(\theta\) for
   1. \(x\) ;
   2. \(y\) .[2]

Let the area of the rectangle be \(A\).

2. Show that \(A=18\sin{2\theta}\).[3]
3. 1. Find \( \frac{\rm{d}A}{\rm{d}\theta}\).
   2. Hence, find the exact value of \(\theta\) which maximizes the area of the rectangle.
   3. Use the second derivative to justify that this value of \(\theta\) does give a maximum.[8]