(i) To find angle AOB, use the tangent formula in the right triangle formed by the radius and the tangent: \(\tan(\frac{x}{2}) = \frac{15}{8} = 1.875\). Solving for \(\frac{x}{2}\), we get \(\frac{x}{2} = 1.081\). Therefore, \(x = 2.16\) radians.

(ii) The perimeter of the shaded region is the sum of the lengths of AT, BT, and the arc AB. The arc length is given by \(r\theta = 8 \times 2.16 = 17.3\) cm. Thus, the perimeter is \(15 + 15 + 17.3 = 47.3\) cm.

(iii) The area of sector AOB is \(\frac{1}{2} r^2 \theta = \frac{1}{2} \times 8^2 \times 2.16 = 69.1\) cm². The area of triangle AOT is \(2 \times \frac{1}{2} \times 8 \times 15 = 120\) cm². The shaded area is \(120 - 69.1 = 50.9\) cm².