(i) To find \(AC\), note that \(AC = OA - OC\). Since \(OC = r \cos \theta\), we have:

\(AC = r - r \cos \theta\).

(ii) For the perimeter of the shaded region ABD:

1. Calculate arc AB: \(\text{arc } AB = \frac{4\pi}{3}\).

2. Calculate arc AD: \(\text{arc } AD = \frac{\pi}{2} \times \text{their } AC = \frac{\pi}{2} \times (4 - 4 \cos \frac{\pi}{3}) = \pi\).

3. Calculate BD: \(BD = 4 \sin \frac{\pi}{3} - \text{their } AC = 2\sqrt{3} - 2\).

4. Add these to find the perimeter: \(\text{Perimeter} = \frac{7\pi}{3} + 2\sqrt{3} - 2\).