9709 P31 - Nov 2016 - Q6
1884
(i) By sketching a suitable pair of graphs, show that the equation \(\csc \frac{1}{2}x = \frac{1}{3}x + 1\) has one root in the interval \(0 < x \leq \pi\).
(ii) Show by calculation that this root lies between 1.4 and 1.6.
(iii) Show that, if a sequence of values in the interval \(0 < x \leq \pi\) given by the iterative formula \(x_{n+1} = 2 \sin^{-1} \left( \frac{3}{x_n + 3} \right)\) converges, then it converges to the root of the equation in part (i).
(iv) Use this iterative formula to calculate the root correct to 3 decimal places. Give the result of each iteration to 5 decimal places.
