Two Triangles From One Sine Value: The Ambiguous Case in a Survey
A surveyor measures a triangular plot ABC. Side a (opposite angle A) is 7.0 m, side b (opposite angle B) is 9.5 m, and angle A is 38°. She computes angle B using the sine law and gets B ≈ 56.3°, then worries because a colleague obtained a different triangle from the same three measurements. The plot must have an area of at least 30 m² to qualify for a zoning variance. The surveyor must decide whether the measurements determine a unique triangle, whether both possible triangles satisfy the area requirement, and how the choice of B affects side c and the area.
1. Use the sine law to find all possible values of angle B, and determine whether the given data produce 0, 1, or 2 triangles.
Read the worked answer
Sine law: sin B / b = sin A / a, so sin B = (b sin A)/a = (9.5 · sin 38°)/7.0.
sin 38° ≈ 0.61566, so 9.5 · 0.61566 ≈ 5.8488; divide by 7.0: sin B ≈ 0.8355.
Since 0 < sin B < 1, there are two candidate angles in (0°, 180°): B₁ = arcsin(0.8355) ≈ 56.7° and B₂ = 180° - 56.7° ≈ 123.3°.
Check the angle sum with A = 38°:
- B₁ = 56.7° gives C₁ = 180 - 38 - 56.7 = 85.3° > 0, valid.
- B₂ = 123.3° gives C₂ = 180 - 38 - 123.3 = 18.7° > 0, valid.
Both are valid, so the data produce two distinct triangles (the ambiguous case). The condition for two triangles is that the altitude h = b sin A ≈ 5.85 m is less than a = 7.0 m and a < b = 9.5 m; both hold. Hence the measurements do not determine a unique plot.
2. Compute side c and the area for each triangle, and decide whether both satisfy the 30 m² requirement.
Read the worked answer
Use the sine law for c: c / sin C = a / sin A, so c = a sin C / sin A = 7.0 sin C / sin 38°.
Triangle 1 (C₁ = 85.3°): sin 85.3° ≈ 0.9966, so c₁ ≈ 7.0 · 0.9966 / 0.61566 ≈ 11.33 m. Area₁ = ½ · a · b · sin C = ½ · 7.0 · 9.5 · sin 85.3° ≈ 33.25 · 0.9966 ≈ 33.1 m².
Triangle 2 (C₂ = 18.7°): sin 18.7° ≈ 0.3206, so c₂ ≈ 7.0 · 0.3206 / 0.61566 ≈ 3.65 m. Area₂ = ½ · 7.0 · 9.5 · sin 18.7° ≈ 33.25 · 0.3206 ≈ 10.7 m².
Only Triangle 1 meets the 30 m² threshold (33.1 m² ≥ 30); Triangle 2 gives about 10.7 m², well below. So the zoning variance is justified only if the plot is the obtuse-C triangle with B ≈ 56.7°. The surveyor cannot rely on the measurements alone; she needs an additional field measurement (for example angle C or side c) to identify which triangle is the real plot.