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In the diagram, \(A\) is a point on the circumference of a circle with centre \(O\) and radius \(r\). A circular arc with centre \(A\) meets the circumference at \(B\) and \(C\). The angle \(O A B\) is equal to \(x\) radians. The shaded region is bounded by \(A B , A C\) and the circular arc with centre \(A\) joining \(B\) and \(C\). The perimeter of the shaded region is equal to half the circumference of the circle.
- Show that \(x = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x } \right)\).
- Verify by calculation that \(x\) lies between 1 and 1.5.
- Use the iterative formula
$$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { \pi } { 4 + 4 x _ { n } } \right)$$
to determine the value of \(x\) correct to 2 decimal places. Give the result of each iteration to 4 decimal places.