Introduction
The SSA configuration is called "ambiguous" because knowing two sides and a non-included angle doesn't uniquely determine a triangle.
Prerequisite Connection
You know the Law of Sines from the previous lesson.
Today's Increment
We're learning to determine how many triangles exist in SSA situations.
Why This Matters
Understanding when a problem has multiple solutions is critical for engineering design.
The Ambiguous Case Analysis
Given angle A, side a (opposite), and side b (adjacent):
No Triangle
If
One Triangle
If
Two Triangles
If and
Two-Triangle Case: The second angle is .
Why Two Triangles? — Watch side "a" swing like a compass
Side "a" swings and hits the base at B₁ or B₂ — two valid triangles!
Worked Examples
Example 1: No Triangle Exists
Given , , .
Answer: No triangle exists
Example 2: One Triangle — Complete Solution
Given , , . Since , at most one triangle exists.
Find Angle B
Find Angle C
Find Side c
Answer:
Example 3: Two Triangles — Complete Solutions
Given , , . Since , check for two triangles.
Find sin B
Find Both Possible B Values
Triangle 1 ()
Triangle 2 ()
Two Valid Triangles:
△1:
△2:
Common Pitfalls
Forgetting to check for two triangles
When , always compute .
Not validating the second solution
Verify for the second triangle to be valid.
Real-World Application
GPS & Triangulation
GPS systems use triangulation that can encounter ambiguous cases. Engineers must verify unique solutions exist.
Practice Quiz
Loading...