Understanding SSA Triangles
The SSA (Side-Side-Angle) case occurs when you know two sides of a triangle and an angle that is NOT between them. This is also known as the "ambiguous case" because the given information may result in zero, one, or two possible triangles.
The Ambiguous Case
No Solution
When side a is too short to form a triangle. This happens when a < b sin(A).
a < b x sin(A)
One Solution (right triangle)
When side a equals the height from vertex B. a = b sin(A) produces exactly one right triangle.
a = b x sin(A)
Two Solutions
When b sin(A) < a < b, there are two possible triangles (the ambiguous case).
b sin(A) < a < b
One Solution
When a ≥ b (and A is acute) or A is obtuse, there is exactly one solution.
a ≥ b
How to Solve SSA Triangles
- Use the Law of Sines to find angle B: sin(B) = b x sin(A) / a.
- Check if sin(B) > 1 (no solution) or sin(B) = 1 (right triangle).
- If sin(B) < 1, find B = arcsin(sin(B)). A second solution may exist: B' = 180 - B.
- Find angle C = 180 - A - B (must be positive for a valid triangle).
- Use the Law of Sines to find side c: c = a x sin(C) / sin(A).
- Calculate area using: Area = (1/2) x a x b x sin(C).