By sketching a suitable pair of graphs, show that the equation
$$\cos x = 2 - 2 x$$
where \(x\) is in radians, has only one root for \(0 \leqslant x \leqslant \frac { 1 } { 2 } \pi\).
Verify by calculation that this root lies between 0.5 and 1 .
Show that, if a sequence of values given by the iterative formula
$$x _ { n + 1 } = 1 - \frac { 1 } { 2 } \cos x _ { n }$$
converges, then it converges to the root of the equation in part (i).
Use this iterative formula, with initial value \(x _ { 1 } = 0.6\), to determine this root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.