Given the problem setup with angles $A = 51^\circ$, side $a = 8.3$, side $b = 9.3$, and the results for the two possible triangles, we are dealing with the ambiguous case of the Law of Sines. This occurs when solving triangles with SSA (two sides and a non-included angle), and there can be either one, two, or no possible solutions.

Let's check the results based on your input:

Smaller $B_1$-value:

• $B_1 = 60.77^\circ$
• $C_1 = 68.23^\circ$
• $c_1 = 10.53$

Larger $B_2$-value:

• $B_2 = 119.23^\circ$
• $C_2 = 9.77^\circ$
• $c_2 = 1.80$

The sum of the angles in a triangle must equal $180^\circ$, and we can confirm that both cases satisfy this condition:

• $A + B_1 + C_1 = 51^\circ + 60.77^\circ + 68.23^\circ = 180^\circ$
• $A + B_2 + C_2 = 51^\circ + 119.23^\circ + 9.77^\circ = 180^\circ$

Both values of $B_1$ and $B_2$ are valid solutions for this ambiguous case, and the side lengths $c_1$ and $c_2$ match the expected results based on the Law of Sines.

Would you like further clarification or help with another aspect of this problem?

5 Follow-up Questions:

1. Would you like to see the steps to calculate angles $B$ and $C$ using the Law of Sines?
2. How is the ambiguous case of the Law of Sines handled in general?
3. Do you need help verifying the side lengths $c_1$ and $c_2$?
4. Are there any other cases where the Law of Sines can produce ambiguous results?
5. Can I assist you with a diagram to visualize both triangles?

Tip:

In the ambiguous case (SSA), always check if two triangles are possible by determining whether the second possible angle for $B$ results in a valid triangle.