By sketching a suitable pair of graphs, show that the equation
$$\sec x = 3 - x$$
where \(x\) is in radians, has only one root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
Verify by calculation that this root lies between 1.0 and 1.2.
Show that this root also satisfies the equation
$$x = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x } \right)$$
Use the iterative formula
$$x _ { n + 1 } = \cos ^ { - 1 } \left( \frac { 1 } { 3 - x _ { n } } \right)$$
with initial value \(x _ { 1 } = 1.1\), to calculate the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.