1. **Problem:** Solve the equation $$12 \sin^2 x - 11 \sin x + 2 = 0$$ for $$\sin x$$. 2. **Formula and approach:** This is a quadratic equation in terms of $$\sin x$$. Let $$y = \sin x$$, then the equation becomes: $$12y^2 - 11y + 2 = 0$$ 3. **Solve the quadratic equation:** Use the quadratic formula: $$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $$a=12$$, $$b=-11$$, and $$c=2$$. 4. **Calculate the discriminant:** $$\Delta = (-11)^2 - 4 \times 12 \times 2 = 121 - 96 = 25$$ 5. **Find the roots:** $$y = \frac{11 \pm \sqrt{25}}{24} = \frac{11 \pm 5}{24}$$ 6. **Evaluate each root:** - $$y_1 = \frac{11 + 5}{24} = \frac{16}{24} = \frac{2}{3} \approx 0.6667$$ - $$y_2 = \frac{11 - 5}{24} = \frac{6}{24} = \frac{1}{4} = 0.25$$ 7. **Interpretation:** Since $$y = \sin x$$, the solutions for $$\sin x$$ are $$\frac{2}{3}$$ and $$\frac{1}{4}$$. **Final answer:** $$\sin x = \frac{2}{3}$$ or $$\sin x = \frac{1}{4}$$.