Navigation and Surveying

Pre-Calculus · Axiom Academy

When you can't walk the distance, you measure the angles — and let the Law of Sines and Law of Cosines finish the triangle. A surveyor needs the distance across a pond she can't cross. She walks a 120 m baseline and sights the far landmark from each end. From those two angles alone, the whole triangle — and every distance in it — falls out. Here are three moves that idea makes. You can measure two angles to the landmark from a known baseline — but not the distances to it. Drag each angle and watch the Law of Sines hand you both distances at once. Two legs and the turn between them A 450 mi flight detours through a fuel stop 280 mi out. Here you know two sides and the angle between them — no opposite angle for the Law of Sines. Drag the turn angle and let the Law of Cosines reach across the gap. One reading, how many answers? Knowing an angle and two sides usually pins a triangle down — but when the known angle is not between the sides, the sightline can sometimes land in two places. Swing the measured length and watch the number of valid triangles flip. Three moves, one idea: two angles and a side → the Law of Sines; two sides and the angle between → the Law of Cosines; and watch for the ambiguous case when the known angle is opposite a given side. It's the same triangulation a ship's navigator, a land surveyor, and the GPS in your pocket all run to find a distance no one can walk.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.