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\includegraphics[max width=\textwidth, alt={}, center]{3c63c42a-2658-4984-93e8-b2a8d18eb37a-14_407_734_267_699} The diagram shows a semicircle with diameter \(A B\), centre \(O\) and radius \(r\). The shaded region is the minor segment on the chord \(A C\) and its area is one third of the area of the semicircle. The angle \(C A B\) is \(\theta\) radians.

1. Show that \(\theta = \frac { 1 } { 3 } ( \pi - 1.5 \sin 2 \theta )\).
2. Verify by calculation that \(0.5 < \theta < 0.7\).
3. Use an iterative formula based on the equation in part (a) to determine \(\theta\) correct to 3 decimal places. Give the result of each iteration to 5 decimal places.