(i) Since arc OC is part of a circle with centre A, the angle CAO is \(\frac{\pi}{3}\) radians.

(ii) The area of sector AOC is given by \(\frac{1}{2} r^2 \times \frac{\pi}{3}\).

The area of triangle ABC is \(\frac{1}{2} (r)(2r) \sin\left(\frac{\pi}{3}\right)\).

Using \(\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}\), the area of triangle ABC becomes \(\frac{1}{2} (2r^2) \frac{\sqrt{3}}{2} = \frac{r^2 \sqrt{3}}{2}\).

The area of the shaded region is the area of triangle ABC minus the area of sector AOC, which is \(\frac{r^2 \sqrt{3}}{2} - \frac{1}{2} r^2 \frac{\pi}{3}\).