You are told that $\int_0^a x \cos\!\left(\tfrac{1}{3}x\right) dx = 3$, with the constant $a$ chosen so that $0 < a < \tfrac{3}{2}\pi$.
(i)[5]
Show that $a$ is a solution of the equation $a = \dfrac{4 - 3\cos\!\left(\tfrac{1}{3}a\right)}{\sin\!\left(\tfrac{1}{3}a\right)}$.
(ii)[2]
Use calculation to confirm that $a$ lies between $2.5$ and $3$.
(iii)[3]
Use an iterative formula derived from the equation in part (i) to find $a$ correct to $3$ decimal places. Show each iteration to $5$ decimal places.
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