Question:

Step 1: To solve the given trigonometric equation $\sin\theta = \frac{7\sin\theta}{11\cos\theta}$, we first need to manipulate the equation to find a relationship between $\sin\theta$ and $\cos\theta$ that can help us determine the possible values of $\theta$. Step 2: We start by cross-multiplying to get $11\cos\theta\sin\theta = 7\sin\theta$, which simplifies to $11\cos\theta = 7$ after dividing both sides by $\sin\theta$, assuming $\sin\theta \neq 0$. Step 3: Rearranging the equation gives us $\cos\theta = \frac{7}{11}$. Step 4: However, the original solution provided does not directly relate to the given equation but instead discusses an inequality $D = 4(1 - \sin\theta)^2 - 4(1 - \sin\theta) \cdot 3\sin\theta \geq 0$. Let's expand and simplify this inequality to understand its implications on $\sin\theta$. Step 5: Expanding the inequality yields $4(1-\sin\theta)(1-\sin\theta - 3\sin\theta) \geq 0$, which simplifies to $4(1-\sin\theta)(1 - 4\sin\theta) \geq 0$. This can be further simplified to $8(1-\sin\theta)(1 - 2\sin\theta) \geq 0$ by factoring out a common factor. Step 6: The inequality $8(1-\sin\theta)(1 - 2\sin\theta) \geq 0$ requires either both factors to be positive or both to be negative. This leads to the conditions $\sin\theta \leq -\frac{1}{2}$ or $\sin\theta \in (\frac{7}{8}, 1]$ after considering the signs of the factors. Step 7: However, the correct interpretation of the inequality should consider the original constraint $-1 \leq \sin\theta \leq 1$. Combining this with the derived conditions, we should focus on the feasible range of $\sin\theta$ that satisfies both the inequality and the constraint. Step 8: Given the misinterpretation in the steps above regarding the direct application of the inequality to the problem, let's refocus on the relationship between $\sin\theta$ and $\cos\theta$ derived from the given equation. The correct approach involves using the equation $\sin\theta = \frac{7\sin\theta}{11\cos\theta}$ to find $\cos\theta = \frac{7}{11}$, which then allows us to find $\sin\theta$ using the identity $\sin^2\theta + \cos^2\theta = 1$. Step 9: Substituting $\cos\theta = \frac{7}{11}$ into the identity $\sin^2\theta + \cos^2\theta = 1$ gives $\sin^2\theta + \left(\frac{7}{11}\right)^2 = 1$. Solving for $\sin^2\theta$ yields $\sin^2\theta = 1 - \left(\frac{7}{11}\right)^2 = 1 - \frac{49}{121} = \frac{72}{121}$. Step 10: Taking the square root of both sides gives $\sin\theta = \pm\sqrt{\frac{72}{121}} = \pm\frac{\sqrt{72}}{11} = \pm\frac{6\sqrt{2}}{11}$. However, the original problem's solution path does not directly follow from the given equation but rather discusses a range for $\sin\theta$. Step 11: Given the confusion in the application of the inequality and the equation, the key insight comes from recognizing that the given equation implies a specific relationship between $\sin\theta$ and $\cos\theta$, which can be used to find the values of $\theta$ that satisfy the equation. Step 12: The final answer is: the correct option is not directly derived through the steps provided but based on the given problem statement and standard trigonometric principles, the solution involves finding $\theta$ such that the given equation holds, considering the constraints on $\sin\theta$ and $\cos\theta$. The final answer should match the option that corresponds to the correct range or value of $\theta$ that satisfies the given equation, which in this context, seems to have been misinterpreted. The final answer is $\boxed{2}$.