By sketching a suitable pair of graphs, show that the equation \(4 - x ^ { 2 } = \sec \frac { 1 } { 2 } x\) has exactly one root in the interval \(0 \leqslant x < \pi\).
Verify by calculation that this root lies between 1 and 2 .
Use the iterative formula \(x _ { n + 1 } = \sqrt { 4 - \sec \frac { 1 } { 2 } x _ { n } }\) to determine the root correct to 2 decimal places. Give the result of each iteration to 4 decimal places.