1. **State the problem:** Calculate the value of the expression $$\frac{16 \sin 19^\circ}{\sin 33^\circ}$$. 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 the sine values:** $$\sin 19^\circ \approx 0.32557$$ $$\sin 33^\circ \approx 0.54464$$ 4. **Substitute the values into the expression:** $$\frac{16 \times 0.32557}{0.54464}$$ 5. **Multiply numerator:** $$16 \times 0.32557 = 5.20912$$ 6. **Divide numerator by denominator:** $$\frac{5.20912}{0.54464}$$ 7. **Simplify the fraction:** $$\frac{\cancel{5.20912}}{\cancel{0.54464}} = 9.56$$ (rounded to two decimal places) **Final answer:** $$\boxed{9.56}$$