First, find the angle ∠AOC using the arc length formula:

Arc length = radius × angle in radians.

Given arc length = 6 cm and radius = 5 cm, we have:

6 = 5 × ∠AOC

∠AOC = \frac{6}{5} \text{ radians} \approx 1.2 \text{ radians}

Next, find the length of AB using the tangent function:

AB = 5 \times \tan(1.2) \approx 12.86 \text{ cm}

Calculate the area of triangle OAB:

\text{Area of } \triangle OAB = \frac{1}{2} \times 5 \times 12.86 \approx 32.15 \text{ cm}^2

Calculate the area of sector OAC:

\text{Area of sector OAC} = \frac{1}{2} \times 5^2 \times 1.2 \approx 15 \text{ cm}^2

Finally, find the area of the shaded region:

\text{Shaded region} = 32.15 - 15 = 17.2 \text{ cm}^2