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To solve this problem using the Law of Sines, we'll follow these steps systematically:

### Given Information:
- \( \angle B = 50^\circ \)
- \( a = 102 \)
- \( b = 47 \)

### Step 1: Apply the Law of Sines
The Law of Sines states:
\[
\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}
\]

We can use the known values to find \( \sin(A) \):
\[
\frac{102}{\sin(A)} = \frac{47}{\sin(50^\circ)}
\]

\[
\sin(A) = \frac{102 \cdot \sin(50^\circ)}{47}
\]

Calculate \( \sin(50^\circ) \approx 0.7660 \):
\[
\sin(A) = \frac{102 \cdot 0.7660}{47} \approx \frac{78.132}{47} \approx 1.662
\]

### Step 2: Check the Validity of \( \sin(A) \)
Since \( \sin(A) \) must be between -1 and 1, \( \sin(A) = 1.662 \) is not possible. This indicates that no triangle exists with the given dimensions.

### Conclusion
Since \( \sin(A) > 1 \), no solution exists for this triangle. Therefore:

- \( \angle A = \text{DNE} \)
- \( \angle C = \text{DNE} \)
- \( c = \text{DNE} \)

Would you like to ask any further questions or want more detailed explanations?

### Additional Questions
1. What is the Law of Sines and how is it used in solving triangles?
2. How can we determine if a solution is valid when using trigonometric functions like sine?
3. Why is it important that the sine of an angle must be between -1 and 1?
4. What would happen if the given dimensions were slightly different?
5. Can you have a triangle with a sine value that equals exactly 1?

### Tip
Always verify the sine value's range when solving triangles. If it exceeds 1 or is less than -1, the triangle cannot exist.

Using the Law of Sines to solve for all possible triangles if ∠B = 50°, a = 102, b = 47. If no answer exists, enter DNE for all answers. Assume ∠A is opposite side a, ∠B is opposite side b, and ∠C is opposite side c.

Learn how to solve a triangle using the Law of Sines when given angle ∠B = 50°, side a = 102, and side b = 47. This example explores the application of trigonometry to check for possible triangles and discusses when a triangle solution does not exist.

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