Bearings are a way to describe direction in navigation using angles measured clockwise from north. They are used by pilots, sailors, hikers, surveyors, and map readers because they give a clear direction of travel from one point to another. Unlike many geometry angles, bearings always start at north and turn clockwise, so a bearing of 090° points east and 180° points south.
This makes bearings a practical link between maps, compass directions, and trigonometry.
Key Facts
- A bearing is measured clockwise from north and is usually written with three digits, such as 045° or 270°.
- Compass directions match key bearings: North = 000° or 360°, East = 090°, South = 180°, West = 270°.
- To convert a bearing b to a standard position angle θ measured counterclockwise from east, use θ = 90° - b if positive, or add 360° if needed.
- For a distance d traveled on bearing b, the east component is x = d sin b and the north component is y = d cos b.
- For a right triangle navigation problem, SOH-CAH-TOA applies: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.
- The reverse bearing is found by adding or subtracting 180°: reverse = b + 180° if b < 180°, and reverse = b - 180° if b ≥ 180°.
Vocabulary
- Bearing
- A bearing is an angle that gives direction, measured clockwise from north.
- Compass rose
- A compass rose is a diagram on a map that shows directions such as north, east, south, and west.
- Standard position angle
- A standard position angle is measured counterclockwise from the positive x-axis, which usually points east on a map.
- Resultant displacement
- Resultant displacement is the straight-line distance and direction from the starting point to the final position.
- Reverse bearing
- A reverse bearing is the direction from the destination back to the starting point.
Common Mistakes to Avoid
- Measuring bearings counterclockwise from east, which is wrong because bearings are measured clockwise from north. Always start at the north line on the compass.
- Writing a bearing as 45° instead of 045°, which can be unclear in navigation. Bearings are commonly written with three digits to avoid confusion.
- Using x = d cos b and y = d sin b for bearing components, which swaps the map directions. Since bearing is measured from north, use east component x = d sin b and north component y = d cos b.
- Forgetting that the reverse bearing differs by 180°, which gives the same path line but opposite travel direction. Add 180° for bearings below 180° and subtract 180° for bearings at least 180°.
Practice Questions
- 1 A boat travels 12 km on a bearing of 060°. Find its east and north components to the nearest tenth of a kilometer.
- 2 A plane flies 80 km east and then 60 km north. Find the straight-line distance from the starting point and the bearing of the final position to the nearest degree.
- 3 A hiker walks from camp to a lake on a bearing of 135°. Explain how to find the bearing from the lake back to camp and describe the compass direction it points.