(i)[2]
By sketching a suitable pair of graphs, show that the equation $\sec x = 3 - \tfrac{1}{2}x^2$, where $x$ is in radians, has a root in the interval $0 < x < \tfrac{1}{2}\pi$.
(ii)[2]
Verify by calculation that this root lies between $1$ and $1.4$.
(iii)[1]
Show that this root also satisfies the equation $x = \cos^{-1}\!\left(\frac{2}{6 - x^2}\right)$.
(iv)[3]
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.