(a) Sketch the graph \(y = \sec x\) and the line \(y = 2 - \frac{1}{2}x\). The intersection point in the interval \(0 \leq x < \frac{1}{2}\pi\) indicates the root. The graphs intersect exactly once in this interval.

(b) Calculate \(\sec(0.8) \approx 1.139\) and \(2 - \frac{1}{2}(0.8) = 1.6\). Calculate \(\sec(1) \approx 1.850\) and \(2 - \frac{1}{2}(1) = 1.5\). Since \(\sec(0.8) < 1.6\) and \(\sec(1) > 1.5\), the root lies between 0.8 and 1.

(c) Using the iterative formula:

\(x_1 = \cos^{-1}\left(\frac{2}{4-0.8}\right) \approx 0.9273\)

\(x_2 = \cos^{-1}\left(\frac{2}{4-0.9273}\right) \approx 0.8834\)

\(x_3 = \cos^{-1}\left(\frac{2}{4-0.8834}\right) \approx 0.8800\)

\(x_4 = \cos^{-1}\left(\frac{2}{4-0.8800}\right) \approx 0.8798\)

The root correct to 2 decimal places is 0.88.