1. **State the problem:** Calculate the value of the expression $$\frac{12 \sin 40^\circ}{\sin 96^\circ}$$. 2. **Formula and rules:** This expression uses the sine function from trigonometry. The sine of an angle in degrees can be found using a calculator or sine tables. 3. **Calculate sine values:** $$\sin 40^\circ \approx 0.6428$$ $$\sin 96^\circ \approx 0.9945$$ 4. **Substitute values into the expression:** $$\frac{12 \times 0.6428}{0.9945}$$ 5. **Multiply numerator:** $$\frac{7.7136}{0.9945}$$ 6. **Divide numerator by denominator:** $$7.7136 \div 0.9945 = \frac{\cancel{7.7136}}{\cancel{0.9945}} \approx 7.756$$ 7. **Final answer:** $$\boxed{7.76}$$ (rounded to two decimal places)