The Ambiguous Sine Rule on a Surveyed Plot
A surveyor is mapping a triangular plot ABC for a landscaping quotation. Standing at corner A, she measures the boundary AB as 12.0 m along a straight fence and records angle BAC = 35°. A second measurement gives BC = 9.0 m, taken across a ditch to the far corner C. The title deed states that the third boundary AC is at least 14 m long, and the client wants the plot area for a turf quote. Back in the office, the assistant enters the three values into the sine rule, obtains a single value for angle BCA, builds a triangle with AC = 4.03 m, and is about to submit a turf quote based on 13.9 m². The surveyor is suspicious: on site the plot is clearly much larger than a small triangle, and the ditch measurement BC is the least reliable of the three because the tape sagged badly in the middle.
1. Using the sine rule, show that the given measurements are ambiguous and find both possible values of angle BCA and of the side AC.
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Sine rule
BC/sin A = AB/sin C, so sin C = (AB sin A)/BC = (12 sin 35°)/9 = (12 × 0.57358)/9 = 0.76477.
Two angles from one sine value
C = 49.89° or C = 180° − 49.89° = 130.11°.
Both are admissible because in each case A + C < 180°, leaving a positive angle at B.
Case 1: C = 49.89°
B = 180° − 35° − 49.89° = 95.11°
AC = (BC sin B)/sin A = (9 sin 95.11°)/sin 35° = (9 × 0.99604)/0.57358 = 15.63 m
Case 2: C = 130.11°
B = 180° − 35° − 130.11° = 14.89°
AC = (9 sin 14.89°)/sin 35° = (9 × 0.25697)/0.57358 = 4.03 m
The assistant silently chose Case 2 - the acute value returned by a calculator - which is exactly the trap.
2. Use the deed to select the correct triangle, then calculate the plot's area and angle ABC.
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Selecting the triangle
The deed requires AC ≥ 14 m. Case 1 gives 15.63 m (acceptable); Case 2 gives 4.03 m (rejected).
So C = 49.89°, B = 95.11°, AC = 15.63 m.
Area
Area = ½ × AB × AC × sin A = ½ × 12 × 15.63 × sin 35°
= 6 × 15.63 × 0.57358 ≈ 53.8 m²
Consistency check with the other pair of sides and the included angle:
½ × AB × BC × sin B = ½ × 12 × 9 × sin 95.11° = 54 × 0.99604 ≈ 53.8 m² ✓
The turf quote based on 13.9 m² underestimates the plot by roughly 40 m², so the assistant's figure must be withdrawn.
3. Explain why these measurements create an ambiguity, and state one extra measurement that would settle the matter without re-measuring the unreliable BC.
Read the worked answer
Why the ambiguity exists
The data are side–side–angle with the angle not included (A, AB, BC). The side opposite the given angle is BC = 9 m. Compare it with the perpendicular height from B to line AC: h = AB sin A = 12 sin 35° = 6.88 m. Since 6.88 < 9 < 12, the side BC can swing into two positions, meeting line AC twice - once on each side of the foot of the perpendicular. Hence two triangles fit the numbers, one with C acute and one with C obtuse.
Extra measurement
Measure angle ABC (or angle ACB). If the angle at B is read as roughly 95°, the obtuse-C triangle is the real one; if it reads about 15°, the small triangle is real. Measuring AC directly to the nearest 0.1 m would also settle it (15.6 m versus 4.0 m), but angles are usually easier to sight across a ditch than a distance, so the field judgement is to measure an angle rather than the sagging tape.