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The diagram shows a semicircle \(A C B\) with centre \(O\) and radius \(r\). The angle \(B O C\) is \(x\) radians. The area of the shaded segment is a quarter of the area of the semicircle.
- Show that \(x\) satisfies the equation
$$x = \frac { 3 } { 4 } \pi - \sin x$$
- This equation has one root. Verify by calculation that the root lies between 1.3 and 1.5.
- Use the iterative formula
$$x _ { n + 1 } = \frac { 3 } { 4 } \pi - \sin x _ { n }$$
to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.