By sketching a suitable pair of graphs, show that the equation
$$\sec x = 3 - \frac { 1 } { 2 } x ^ { 2 }$$
where \(x\) is in radians, has a root in the interval \(0 < x < \frac { 1 } { 2 } \pi\).
Verify by calculation that this root lies between 1 and 1.4.
Show that this root also satisfies the equation
$$x = \cos ^ { - 1 } \left( \frac { 2 } { 6 - x ^ { 2 } } \right)$$
Use an iterative formula based on the equation in part (iii) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.