The ambiguous case of the law of sines, explained
Why solving for an angle from a side can give two triangles, how to spot which situations are ambiguous, and how this tool handles it.
Where the ambiguity comes from
When you know two sides and an angle that is not between them, the arrangement is called side-side-angle. The law of sines finds the unknown angle with an inverse sine, and the sine function returns the same value for an angle and its supplement. An angle of 45 degrees and one of 135 degrees share the same sine, so the equation alone cannot tell which triangle you meant. That single fact is the whole reason the case is called ambiguous.
How many triangles are possible
The side-side-angle setup can produce zero, one or two valid triangles. If the given side opposite the unknown angle is too short to reach the base, no triangle closes and the sine ratio exceeds one. If it is exactly long enough, there is one right-angled triangle. If it is longer than that but shorter than the other given side, both the acute and the obtuse angle can form a genuine triangle, giving two answers. Knowing which of these you are in usually comes from the physical context of the problem.
What this calculator reports
This tool always returns the acute solution when you solve for an angle from a side, because that branch is the more common one and it is the answer most textbooks expect first. When the acute angle plus angle A already reach 180 degrees, the tool reports an error instead of a broken triangle. If your real triangle is obtuse at B, take the value shown and subtract it from 180 to get the second solution, then recompute angle C as 180 minus A minus that obtuse angle.
Avoiding the trap in practice
The cleanest way to sidestep ambiguity is to solve for a side rather than an angle whenever you can, because finding a side from two angles has only one answer. If you must find an angle, sketch the triangle roughly to scale first so you can see whether the answer should look acute or obtuse. That quick sketch turns a confusing algebraic result into an obvious choice between the two candidate triangles.