Solve oblique triangles using the Law of Sines (a/sin A = b/sin B = c/sin C). Handles AAS, ASA, and the SSA ambiguous case — including 0, 1, or 2 triangle solutions. Get a full step-by-step solution and a labeled diagram.
Solve a triangle above to see the step-by-step solution using the Law of Sines.
a / sin A = b / sin B = c / sin C
The Law of Sines states that the ratio of any side to the sine of its opposite angle is constant for a given triangle. Use it whenever you know an angle and the side opposite it, plus one other piece of information.
| Case | Given | Strategy |
|---|---|---|
| AAS | A, B, a |
C = 180° − A − B, then find b and c |
| ASA | A, B, c |
C = 180° − A − B, then find a and b |
| SSA | a, b, A |
Ambiguous — use the height test first |
The Law of Sines does not apply directly to SSS or SAS — those require the Law of Cosines.
h = b · sin A
When given sides a and b with angle A opposite to a, the side a can "swing" and hit the base in 0, 1, or 2 places. The height h = b sin A is the minimum length needed for a to reach the base.
| Condition | Result | Notes |
|---|---|---|
| a < h | 0 triangles | side a is too short to reach the base |
| a = h | 1 right triangle | a exactly reaches — right angle at B |
| h < a < b | 2 triangles | B₁ = arcsin(…), B₂ = 180° − B₁ |
| a ≥ b | 1 triangle | the obtuse alternative would make angle sum > 180° |
When there are 2 solutions, find both values of B and build each triangle independently using the Law of Sines.
A live tutor can walk you through the Law of Sines, the ambiguous case, and the Law of Cosines — with personalized practice problems tailored to your course.