Trigonometry · Intermediate · 11 min
The ambiguous triangle
Two sides and a non-included angle may determine zero, one, or two triangles.
WHAT HAPPENS IF…With A=30°, b=2, and a=1.5, how many triangles are possible?
Loading the mathematical model…
Place the known side b at angle A to the positive base ray. The remaining vertex must be distance a from its endpoint, so it lies on a circle. The intersections with the positive ray count the valid triangles. For acute A, compare a with h and b; tangent and degenerate cases must be handled explicitly.
A detail that matters
The sine rule can give supplementary candidate angles; both must satisfy a valid triangle.
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