The Lighthouse Bearing Problem
A trainee navigator on a training ship is given a chart exercise. From the ship's position, a lighthouse bears 040° (true). After the ship steams on a steady course for some time, the same lighthouse bears 310° (true). The navigator knows the ship travelled 12 nautical miles between the two bearings. A senior rating claims the ship must have been heading due north the whole time. The trainee must test this claim using the geometry of the two bearings and decide whether a due-north course is consistent with the observations, and what the resulting distance off the lighthouse was at the second bearing.
1. Explain why a due-north course is or is not consistent with the two bearings.
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If the ship steams due north, the bearing to a fixed object on the port or starboard side changes smoothly as the ship passes it. A bearing swinging from 040° to 310° is a change of 90° through north (040° → 000° → 310°), which is geometrically possible on a northerly course if the lighthouse lies to the east and the ship passes to its west. So the claim is not ruled out by the bearings alone; the bearings are consistent with a northerly course, but consistency is not proof - other courses could produce the same swing. The trainee should say the claim is possible but unproven from bearings alone.
2. Given the 12 nm run and the 90° bearing swing, find the perpendicular distance off the lighthouse at the second bearing.
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The bearing change is 90°, so the triangle formed by the two ship positions and the lighthouse is right-angled at the lighthouse. The run of 12 nm is the hypotenuse. At the second bearing (310°), the angle between the ship's track and the line to the lighthouse can be found: the first bearing 040° and second 310° differ by 90°, so the two sight lines are perpendicular. In the right triangle, the distance off at the second bearing is the side opposite the angle at the first position. Using the right-triangle relation with a 90° apex and a 12 nm hypotenuse, the two legs are 12·sin and 12·cos of the base angles. With a symmetric 45°/45° split (the two sight lines perpendicular and the run bisecting), each leg = 12 × sin 45° ≈ 12 × 0.707 ≈ 8.5 nm.
3. State one reason the trainee should not rely on bearings alone to confirm the course.
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Bearings give relative geometry only; many combinations of course and speed can reproduce the same pair of bearings. To confirm a due-north course the navigator needs an independent check - a compass heading log, a fix from two or more objects, or a known speed and elapsed time. The correct professional judgement is: bearings are consistent with the claim but insufficient to prove it, so corroborate with a second source.