The diagram shows a trapezium $$\(A B C D\)$$ in which $$\(A D=B C=r\)$$ and $$\(A B=2 r\)$$. The acute angles $$\(B A D\)$$ and $$\(A B C\)$$ are both equal to $$\(x\)$$ radians. Circular arcs of radius $$\(r\)$$ with centres $$\(A\)$$ and $$\(B\)$$ meet at $$\(M\)$$, the midpoint of $$\(A B\)$$. Given that the sum of the areas of the shaded sectors is $$\(90 \%\)$$ of the area of the trapezium, show that $$\(x\)$$ satisfies the equation $$\(x=0.9(2-\cos x) \sin x\)$$. ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

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