Ambiguous Case (SSA)

Trigonometry · Axiom Academy

LESSON The Ambiguous Case (SSA) When two sides and an angle opposite one of them are given, the number of possible triangles isn't always clear. The problem starts when we use the Law of Sines to find angle B. We know the angle A and want to find B, but the situation isn't straightforward because side a can potentially "swing" to meet the base in different ways. Depending on the relationship between the sides and angle, we can get three dramatically different outcomes: 3. The Swinging Side Demonstration Let's see how side a "swings" from vertex C to potentially meet the base. Watch how the circle of radius a centered at C intersects the base in different ways depending on the length of a: No solution: If a < h (side too short) One solution: If a = h (perpendicular) or a ≥ b (long enough) Two solutions: If h < a < b (the ambiguous case) When we use the Law of Sines to find angle B, we get sin(B) = (b · sin(A)) / a. But remember: if sin(B) = k, then B could be either arcsin(k) or 180° - arcsin(k). Calculate h = 20 · sin(40°) ≈ 12.86 Since 12.86 < 15 < 20, we have TWO triangles sin(B) = (20 · sin(40°)) / 15 ≈ 0.857 B₁ ≈ 59° and B₂ ≈ 121° (both valid!)

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