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The problem features a triangle with a given side length of \(11\sqrt{6}\), and two angles \(45^\circ\) and \(60^\circ\). The objective is to determine the length \(x\) of the side opposite the \(60^\circ\) angle.

Let's solve this step by step:

1. **Identify the given elements and the type of triangle**:  
   - The triangle appears to be a right triangle with angles \(45^\circ\), \(60^\circ\), and \(90^\circ\).
   - The given side \(11\sqrt{6}\) is opposite the \(45^\circ\) angle.

2. **Use the sine and cosine rule**:  
   - For the angle \(45^\circ\), the side opposite is the given \(11\sqrt{6}\).
   - The ratio of the sides in a \(45^\circ-45^\circ-90^\circ\) triangle is \(1:1:\sqrt{2}\).

3. **Determine the other side using trigonometric identities**:
   - We can use the sine or cosine for \(60^\circ\).
   - For a right triangle:
     \[
     \text{Opposite to } 60^\circ = x, \quad \text{Adjacent to } 60^\circ = 11\sqrt{6}
     \]
   - By applying the sine rule or cosine rule directly to this triangle:
     \[
     \sin(60^\circ) = \frac{x}{11\sqrt{6}}
     \]
     \[
     \sin(60^\circ) = \frac{\sqrt{3}}{2}
     \]

4. **Solve for \(x\)**:
   \[
   x = 11\sqrt{6} \times \frac{\sqrt{3}}{2} = \frac{11\sqrt{18}}{2} = 11\sqrt{3} \times \sqrt{6} = 11\sqrt{18} = 33\sqrt{2}
   \]

So, the correct answer is **\(33\sqrt{2}\)**, which matches option **B**.

Would you like further details or have any questions? Here are some related questions you might explore:

1. How can we derive the sine and cosine rules?
2. What is the significance of using trigonometric identities in right triangles?
3. How does the \(45^\circ-45^\circ-90^\circ\) triangle differ from other triangles?
4. What are the other methods for solving right triangles?
5. How do you determine when to use the sine, cosine, or tangent function?
6. Can this problem be solved using a different approach?
7. What is the relationship between the angles and sides in a right triangle?
8. How do angles in triangles determine the properties of the sides?

**Tip:** When solving triangles, always check if a simple geometric rule or identity can directly provide the answer before resorting to more complex calculations.

Discover how to calculate the length of a triangle's side opposite a 60° angle. Detailed step-by-step solution using trigonometric identities and rules. Suitable for advanced high school students studying trigonometry and right triangles.

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