We are given a triangle with the following information:

• Angle $A = 58^\circ$
• Side $a = 4.5$
• Side $b = 12.8$

Using the Law of Sines:

$\frac{a}{\sin(A)} = \frac{b}{\sin(B)}$

First, we can rearrange the formula to solve for $\sin(B)$:

$\sin(B) = \frac{b \cdot \sin(A)}{a}$

Now, substituting the values:

$\sin(B) = \frac{12.8 \cdot \sin(58^\circ)}{4.5}$

We need to calculate $\sin(58^\circ)$. Using a calculator:

$\sin(58^\circ) \approx 0.848$

Now substitute this value back into the equation:

$\sin(B) = \frac{12.8 \cdot 0.848}{4.5} \approx \frac{10.8624}{4.5} \approx 2.4149$

Since the sine of an angle cannot be greater than 1, this implies that no solution exists for angle $B$. Therefore, the triangle with the given values does not exist.

This is due to the fact that the ratio of sides $a$ and $b$, with the angle between them, results in an impossible sine value (greater than 1), meaning the given configuration cannot form a valid triangle.

Summary:

The triangle with $A = 58^\circ$, $a = 4.5$, and $b = 12.8$ does not exist because the Law of Sines gives an invalid sine value for $B$.

Would you like me to explore any other triangles or details on how to solve similar problems? Here are a few related questions for you to consider:

1. What happens when the sine of an angle exceeds 1?
2. How do you determine the number of possible triangles given two sides and a non-included angle (SSA)?
3. What is the ambiguous case in the Law of Sines, and how does it affect triangle solutions?
4. How do you solve for the other angles in a triangle if only two sides and the included angle are known?
5. What is the Law of Cosines and how does it compare with the Law of Sines in triangle problems?

Tip: Always check if the sine value you obtain is within the valid range (−1 to 1) when using the Law of Sines to avoid impossible triangles.