Bearings and Solving Triangle Problems
Simple Explanation
A bearing gives a direction as a three-figure angle (000° to 360°) measured CLOCKWISE from north. Bearing problems typically describe a journey using two or more bearings and distances, forming a triangle that can then be solved using the Law of Sines or Law of Cosines.
Why Do We Need It?
Bearings are the standard way distances and directions are described in navigation, and solving the resulting triangles is a direct, practical application of everything else in this chapter.
See It
Point A with a dashed arrow pointing north, and a solid arrow to point B, with the clockwise angle from north to AB marked as 60 degrees
Formula
The Law of Cosines
c² = a² + b² − 2ab·cos C
A generalization of the Pythagorean theorem to any triangle — gives the length of a side from the other two sides and the angle between them, or an angle from all three sides.
- a, b, c
- — the side lengths of the triangle
- C
- — the angle opposite side c, included between sides a and b
When to use it: Whenever you know two sides and the included angle (SAS), or all three sides (SSS), of any triangle.
Worked Example
Solve a two-leg bearing problem
A ship sails from port A on a bearing of 065° for 40 km to point B, then changes course to a bearing of 155° and sails 30 km to point C. Find the distance AC.
Why Does This Work?
Converting the bearings into a single interior angle of the triangle at the shared vertex B is what turns the navigation description into an ordinary triangle problem — from there, the Law of Cosines (or Law of Sines, depending on what's known) solves it exactly as with any other triangle.
Real-Life Example
Marine and aviation navigation
A ship or aircraft needs to know the direct distance and bearing to a destination, after sailing or flying multiple legs on different bearings.
This is exactly the bearings-and-triangle-solving technique used in real marine and aviation navigation and flight planning.
Practice
A hiker walks on a bearing of 040° from point X. What is the bearing from the hiker's new position back to X?
HardCommon mistake
Forgetting that bearings are measured clockwise from NORTH, not from the positive x-axis (east) as in standard angle notation — bearings and standard angles need converting between each other, not treating as the same thing.
Quick Review
- A bearing is a three-figure angle (000°-360°), measured clockwise from north.
- The reverse bearing is the original bearing ± 180°.
- Convert the bearing description into a triangle's interior angle, then apply the Law of Sines/Cosines.