Ambiguous case

Definition

The ambiguous case occurs when solving for a triangle using the Law of Sines and given two sides and a non-included angle (SSA). This situation can result in zero, one, or two possible triangles.

5 Must Know Facts For Your Next Test

1. The ambiguous case arises with SSA (Side-Side-Angle) configurations.
2. To determine the number of possible triangles, one must compare the given side opposite the known angle to the height of the triangle.
3. If \(a < h\), where \(h = b \sin A\), no triangle exists.
4. If \(a = h\), exactly one right triangle exists.
5. If \(a > h\), there can be either one or two possible triangles depending on whether \(a \geq b\) or not.

Review Questions

• What conditions lead to the ambiguous case in solving triangles?
• How do you determine if no triangle, one triangle, or two triangles exist?
• Given values for sides and angles in an SSA configuration, can you identify how many potential solutions there are?

Related terms

Law of Sines: A fundamental relation in trigonometry that states $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ for any triangle.

SSA Configuration: A triangle scenario where two sides and a non-included angle are known.

Triangle Height: $h = b \sin A$, used to determine if zero, one, or two triangles are possible in an SSA scenario.