When solving SSA (Side-Side-Angle) triangles using the law of sines, it's important to recognize that this scenario can lead to three different outcomes: no solution, one solution, or two solutions. This ambiguity arises because the angle given is not between the two sides, making it less straightforward than other triangle types.

To begin solving an SSA triangle, start by applying the law of sines, which states:

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

In this equation, \(A\) and \(B\) are the angles opposite sides \(a\) and \(b\), respectively. For example, if you know \(a = 6\), \(b = 8\), and \(A = 41^\circ\), you can set up the equation to find \(\sin B\):

$$\sin B = \frac{b \cdot \sin A}{a} = \frac{8 \cdot \sin(41^\circ)}{6}$$

Calculating this gives you a value for \(\sin B\). If this value exceeds 1, there is no solution, as the sine function only ranges from -1 to 1. If the value is less than or equal to 1, you can proceed to find angle \(B\) using the inverse sine function:

$$B = \sin^{-1}(0.875)$$

This calculation yields two potential angles for \(B\): the first angle directly from the inverse sine, and the second angle calculated as:

$$B_2 = 180^\circ - B_1$$

Next, you must check the sum of the angles. If the sum of \(A\) and \(B_2\) exceeds 180 degrees, then only one triangle is possible. If it is less than 180 degrees, both triangles are valid solutions.

For the first triangle, you can find the third angle \(C\) using the angle sum property:

$$C = 180^\circ - (A + B_1)$$

For the second triangle, use:

$$C_2 = 180^\circ - (A + B_2)$$

In summary, solving SSA triangles involves careful application of the law of sines, consideration of the possible angles derived from the inverse sine function, and verification of the angle sums to determine the number of valid triangles. This methodical approach allows you to navigate the ambiguity inherent in SSA cases effectively.