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.