Ambiguous case

Math

Definition

A situation in triangle solving (SSA — two sides and a non-included angle) where the given information can produce zero, one, or two valid triangles. This occurs when using the Law of Sines with the arrangement side-side-angle.

Examples

  • Given a = 8, b = 12, and angle A = 30°, there are two possible triangles because the side opposite the given angle is shorter than the other given side but longer than a·sin(A)
  • If the side opposite the given angle is shorter than the height (a < b·sin A), no triangle exists
Key Fact

SSA can give 0, 1, or 2 triangles — check if a < b·sin(A), a = b·sin(A), or a > b·sin(A)

Related Terms

Law of sines Law of cosines Solving right triangles Triangles

Study This Concept

Practice ambiguous case with free review games in these units:

Trigonometry Unit 7: Law of Sines and Cosines