(a)[3]
Using that the total area of the shaded sectors is $90\%$ of the area of the trapezium, prove that $x$ satisfies $x = 0.9(2 - \cos x)\sin x$.
(b)[2]
Check by calculation that $x$ is between $0.5$ and $0.7$.
(c)[2]
Show that, if a sequence of values in the interval $0 < x < \frac{1}{2}\pi$ produced by $x_{n+1} = \cos^{-1}\!\left(2 - \frac{x_n}{0.9\sin x_n}\right)$ converges, then its limit is the root of the equation in part (a).
(d)[3]
Apply this iterative formula to find $x$ correct to $2$ decimal places. State each iteration to $4$ decimal places.