The diagram shows a trapezium $$\(A B C D\)$$ in which $$\(A D=B C=r\)$$ and $$\(A B=2 r\)$$. The acute angles $$\(B A D\)$$ and $$\(A B C\)$$ are both equal to $$\(x\)$$ radians. Circular arcs of radius $$\(r\)$$ with centres $$\(A\)$$ and $$\(B\)$$ meet at $$\(M\)$$, the midpoint of $$\(A B\)$$. Show that if a sequence of values in the interval $$\(0<x<\frac{1}{2} \pi\)$$ given by the iterative formula $$\[ x_{n+1}=\cos ^{-1}\left(2-\frac{x_{n}}{0.9 \sin x_{n}}\right) \]$$ converges, then it converges to the root of the equation in part (a). ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

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