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The figure shows the circle with centre \(O\) and radius \(r\).

Points \(P\), \(R\) and \(Q\) are on the circumference, \(\angle\,POQ = 2\theta\), for \(0 < \theta < \frac{\pi}{2}\).

1. Use the law of cosines to show that \(PQ = 2r\,\sin\,\theta\).
2. Let \(l\) be the length of arc \(PRQ\). Given that \(1.3PQ - l = 0\), find the value of \(\theta\).
3. Consider the function \(f(\theta) = 2.6\,\sin\,\theta - 2\theta\), for \(0 < \theta < \frac{\pi}{2}\).
   1. Sketch the graph of \(f\).
   2. Give the root of \(f(\theta) = 0\).
4. Use the curve \(f\) to find the values of \(\theta\) for which \(l < 1.3\,PQ\).