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To solve for \( BC \) in the given triangle using the Law of Sines, we can follow these steps:

### Step-by-Step Solution

1. **Identify the Given Information:**
   - Angle \( \angle A = 61^\circ \)
   - Angle \( \angle C = 30^\circ \)
   - Side \( AB = 8 \) (opposite to angle \( C \))

2. **Calculate the Third Angle:**
   Since the sum of the angles in a triangle is \( 180^\circ \):
   \[
   \angle B = 180^\circ - \angle A - \angle C = 180^\circ - 61^\circ - 30^\circ = 89^\circ
   \]

3. **Apply the Law of Sines:**
   The Law of Sines states:
   \[
   \frac{BC}{\sin(A)} = \frac{AB}{\sin(B)}
   \]
   Therefore:
   \[
   BC = AB \times \frac{\sin(A)}{\sin(B)}
   \]

4. **Plug in the Values:**
   \[
   BC = 8 \times \frac{\sin(61^\circ)}{\sin(89^\circ)}
   \]

5. **Calculate the Sine Values:**
   - \(\sin(61^\circ) \approx 0.8746\)
   - \(\sin(89^\circ) \approx 0.9998\)

6. **Compute \( BC \):**
   \[
   BC = 8 \times \frac{0.8746}{0.9998} \approx 8 \times 0.8748 \approx 6.998
   \]

### Final Answer
\[
BC \approx 7
\]

Would you like a detailed explanation or have any questions about this solution?

### Related Questions

1. How do you derive the Law of Sines?
2. Can you solve for \( AC \) using the Law of Sines?
3. What is the Law of Cosines and when should it be used instead of the Law of Sines?
4. How do you verify the accuracy of trigonometric calculations?
5. Can you solve for angle \( \angle B \) using the given sides?
6. What are the common trigonometric functions and their applications?
7. How do you find the area of a triangle given two angles and one side?
8. How can you apply the Law of Sines to non-right triangles?

### Tip

When using trigonometric functions, always ensure your calculator is set to the correct mode (degrees or radians) corresponding to the angle measurements given in the problem.

Learn how to solve for BC in a triangle using the Law of Sines with a detailed step-by-step solution. This problem covers trigonometric concepts and applications, including calculating side lengths in triangles. Suitable for high school students.

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