Answer: \(\theta=0.8\) radians; shaded area \(=16.5\text{ cm}^2\) approximately.

The arc length formula is

\(s=r\theta.\)

Here \(s=9.6\) and \(r=12\), so

\(9.6=12\theta,\qquad \theta=\boxed{0.8}.\)

The shaded region is the area of triangle \(OAB\) minus the area of sector \(OAC\).

Since \(\tan\theta=\frac{AB}{OA}\),

\(AB=12\tan0.8.\)

So the area of triangle \(OAB\) is

\(\frac12(12)(12\tan0.8)=72\tan0.8.\)

The area of sector \(OAC\) is

\(\frac12r^2\theta=\frac12(12)^2(0.8)=57.6.\)

Therefore the shaded area is

\(72\tan0.8-57.6=16.5339\ldots\)

Thus the area is

\(\boxed{16.5\text{ cm}^2}\)

to 3 significant figures.