What is the ambiguous case in trigonometry?
It arises when you know two sides and a non-included angle (SSA). The given side opposite the angle can sometimes form two different valid triangles, one triangle, or none at all.
// maths › Non-Right-Angle Trigonometry
Ambiguous case (SSA)
Resolve the ambiguous SSA case where two sides and a non-included angle may give no triangle, one triangle, or two different triangles, in degrees or radians, with both valid triangles drawn side by side and an explanation of which case applies and why.
sin B = b·sin A / a → B or 180°−B may both be valid
Angle unit
Frequently asked questions
Why can there be two triangles?
The sine rule gives sin B, and both an acute angle and its obtuse supplement (180° − B) have the same sine. If both keep the angle sum under 180°, both produce a genuine triangle.
Can you give a worked example?
With a = 6, b = 8, and A = 30°: sin B = 8·sin 30° / 6 ≈ 0.667, giving B ≈ 41.8° or B ≈ 138.2°. Both fit, so there are two triangles. In radians, A = 30° ≈ 0.5236 rad; the Degrees/Radians toggle switches the input and every angle in the working.
How do I know how many triangles there are?
If the computed sine exceeds 1, no triangle exists. If only the acute angle keeps the total under 180°, there is one. If both the acute and obtuse options work, there are two.
Where is this used in real life?
Surveyors, navigators, and engineers must recognise when measurements leave a position ambiguous, so they take an extra reading to decide which of the two possible triangles is the real one.