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We are tasked with solving a triangle using the Law of Sines. The given values are:

- \( A = 34.9^\circ \)
- \( a = 27 \)
- \( b = 32 \)

To solve this, we'll apply the Law of Sines:

\[
\frac{\sin A}{a} = \frac{\sin B}{b}
\]

We will calculate the value of angle \( B \), and depending on the results, check if there are one or two possible solutions. Then, we can compute angle \( C \) using the fact that the sum of angles in a triangle is \( 180^\circ \) and finally calculate side \( c \) using the Law of Sines again.

Let me start calculating these values.Based on the calculations:

- For **Solution 1** (smaller angle \( B_1 \)):
  - \( B_1 = 42.7^\circ \)
  - \( C_1 = 102.4^\circ \)
  - \( c_1 = 46 \) (rounded)

- For **Solution 2** (larger angle \( B_2 \)):
  - \( B_2 = 137.3^\circ \)
  - \( C_2 = 7.8^\circ \)
  - \( c_2 = 6.4 \) (rounded)

This means there are **two possible solutions** for the triangle, so the correct choice is **B**. 

Would you like further clarification or additional details?

Here are 5 related questions to explore:
1. What happens if the sine of an angle is greater than 1?
2. How does the Law of Sines help in solving non-right triangles?
3. Can the Law of Sines be used in all triangles? Why or why not?
4. How do you handle cases where there are two possible solutions for a triangle?
5. What are some real-world applications of the Law of Sines?

**Tip:** Always check if the sine value is between -1 and 1 to ensure valid angle calculations when using the Law of Sines.

Use the Law of Sines to solve the triangle, if possible. Given A = 34.9°, a = 27, b = 32.

Solve a triangle using the Law of Sines for A = 34.9°, a = 27, b = 32. The problem explores the ambiguous case, yielding two possible solutions for angles B and C and side c.

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