Introduction
When given two sides and an angle opposite one of them (SSA), the Law of Sines can produce zero, one, or two valid triangles.
Past Knowledge
You know the Law of Sines from the previous lesson.
Today's Goal
We're learning to determine how many triangles exist in SSA situations.
Future Success
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
Practice Quiz
Loading...