(i) The perimeter of the rectangle is \(2(3a + a) = 8a\). Half of this is \(4a\).

The perimeter of the sector AMN is \(r \cdot x + 2r\).

Equating the two perimeters: \(r \cdot x + 2r = 4a\).

Factor out \(r\): \(r(x + 2) = 4a\).

Since \(r = a \csc x\), substitute to get \(a \csc x (x + 2) = 4a\).

Cancel \(a\): \(\csc x (x + 2) = 4\).

\(\csc x = \frac{1}{\sin x}\), so \(\frac{1}{\sin x} (x + 2) = 4\).

Rearrange to get \(\sin x = \frac{1}{4}(2 + x)\).

(ii) Using the iterative formula \(x_{n+1} = \sin^{-1}\left(\frac{2 + x_n}{4}\right)\):

Start with \(x_1 = 0.8\).

\(x_2 = \sin^{-1}\left(\frac{2 + 0.8}{4}\right) = \sin^{-1}(0.7) \approx 0.7754\).

\(x_3 = \sin^{-1}\left(\frac{2 + 0.7754}{4}\right) = \sin^{-1}(0.69385) \approx 0.7672\).

\(x_4 = \sin^{-1}\left(\frac{2 + 0.7672}{4}\right) = \sin^{-1}(0.6918) \approx 0.7650\).

\(x_5 = \sin^{-1}\left(\frac{2 + 0.7650}{4}\right) = \sin^{-1}(0.69125) \approx 0.7643\).

The root correct to 2 decimal places is \(0.76\).