In this example, we explore a triangle with sides \( a = 1 \), \( b = 4 \), and angle \( A = 30^\circ \). This configuration represents an SSA (Side-Side-Angle) triangle, which can lead to ambiguous cases where a triangle may not exist. To analyze this, we can use the Law of Sines, which states:

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

Here, we aim to find angle \( B \). Rearranging the equation gives us:

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

Substituting the known values, we have:

\[\sin B = \frac{4 \cdot \sin(30^\circ)}{1}\]

Since \( \sin(30^\circ) = \frac{1}{2} \), this simplifies to:

\[\sin B = \frac{4 \cdot \frac{1}{2}}{1} = 2\]

However, the sine of an angle cannot exceed 1, indicating that there is no possible angle \( B \) that satisfies this equation. Therefore, we conclude that a triangle cannot be formed with the given dimensions.

To visualize this scenario, consider the angle \( A \) of \( 30^\circ \) and side \( b = 4 \). The side \( a = 1 \) is insufficient to connect to the endpoint of side \( b \), regardless of how it is positioned. This geometric limitation confirms that no triangle can exist under these conditions, leading us to the final conclusion that there is no solution for this triangle configuration.