Compass & True Bearings
Master compass and true bearings for NSW Year 12 Mathematics Standard 2. A true (three-figure) bearing is measured clockwise from north — like \(205^\circ\text{T}\) — while a compass bearing such as \(\text{N}35^\circ\text{E}\) turns from north or south toward east or west.
You will learn to convert between the two forms, find the back (reverse) bearing between two points, and use bearings in right-angled-triangle distance and direction problems — core Standard 2 skills for navigation, orienteering and surveying.
Theory
Bearings describe direction as an angle. This Year 12 Standard 2 (NSW) guide shows how to measure a true (three-figure) bearing clockwise from north, convert between compass and true bearings, find a back (reverse) bearing, and use bearings in right-angled-triangle distance and direction problems.
A bearing gives the direction of one point from another as an angle. In NSW Year 12 Mathematics Standard 2 you use two kinds and convert between them.
A true (three-figure) bearing is the angle turned clockwise from north, always written with three digits and a T — for example \(030^\circ\text{T}\) or \(205^\circ\text{T}\). North is \(000^\circ\text{T}\), east \(090^\circ\), south \(180^\circ\) and west \(270^\circ\).
A compass bearing starts at north or south and turns a stated angle toward east or west, such as \(\text{N}35^\circ\text{E}\) or \(\text{S}25^\circ\text{W}\). The back (reverse) bearing — the return direction between two points — differs by \(180^\circ\).
Step around the compass to convert between the two forms:
The back (reverse) bearing differs by \(180^\circ\):
For a distance–direction problem, draw a right-angled triangle and let \(\theta\) be the angle from the north–south line:
How to work with bearings
- Face north and measure the true bearing clockwise; write it with three figures and a T.
- Compass → true: start at \(000^\circ\) (N) or \(180^\circ\) (S), then add or subtract the turn toward E or W.
- True → compass: name the nearer of N or S, then state the acute angle toward E or W.
- Back bearing: add \(180^\circ\) if the bearing is under \(180^\circ\), or subtract \(180^\circ\) if it is over — keep the answer in \(000^\circ\)–\(360^\circ\).
- Distance and direction: draw the right-angled triangle and use \(d\cos\theta\) and \(d\sin\theta\), or \(\tan\theta\) to find an unknown bearing.
Start at due south \((180^\circ)\) and turn \(25^\circ\) further clockwise toward west.
| \(\text{S}\) | \(=\) | \(180^\circ\text{T}\) |
| \(180^\circ + 25^\circ\) | \(=\) | \(205^\circ\) |
| \(\therefore\ \text{bearing}\) | \(=\) | \(205^\circ\text{T}\) |
\(295^\circ\) is between west \((270^\circ)\) and north \((360^\circ)\), so it is in the NW quadrant.
| \(360^\circ - 295^\circ\) | \(=\) | \(65^\circ\) |
| \(\text{from N, toward W}\) | \(\) | \(\) |
| \(\therefore\ \text{bearing}\) | \(=\) | \(\text{N}65^\circ\text{W}\) |
The return direction differs by \(180^\circ\); since \(108^\circ<180^\circ\), add \(180^\circ\).
| \(108^\circ + 180^\circ\) | \(=\) | \(288^\circ\) |
| \(\therefore\ B\text{ from }L\) | \(=\) | \(288^\circ\text{T}\) |
Draw a right-angled triangle; the angle from north is \(50^\circ\).
| \(\text{North}\) | \(=\) | \(20\cos 50^\circ = 12.9\text{ km}\) |
| \(\text{East}\) | \(=\) | \(20\sin 50^\circ = 15.3\text{ km}\) |
The ship is 12.9 km north and 15.3 km east of its start.
Common pitfalls
Frequently asked questions
What is the difference between a true bearing and a compass bearing?
A true (three-figure) bearing is the angle measured clockwise from north, written with three digits and a T, such as 205 degrees T. A compass bearing starts at north or south and turns toward east or west, such as S25 degrees W. They describe the same direction two different ways.
How do you write a bearing with three figures?
Measure the angle clockwise from north and pad it to three digits with leading zeros. Thirty degrees becomes 030 degrees T and five degrees becomes 005 degrees T. This keeps every true bearing between 000 and 360 degrees.
How do you convert a compass bearing to a true bearing?
Start at 000 degrees for north or 180 degrees for south, then add or subtract the compass angle depending on whether you turn toward east or west. For example S25 degrees W is 180 plus 25, which is 205 degrees T.
What is a back bearing and how do you find it?
A back bearing (or reverse bearing) is the return direction between two points, and it differs by 180 degrees. Add 180 degrees if the original bearing is under 180, or subtract 180 if it is over, so the answer stays between 000 and 360 degrees.
How do you use a bearing in a right-angled triangle problem?
Draw the direction as a ray, then drop a right-angled triangle with legs pointing north-south and east-west. The distance travelled is the hypotenuse, so the north or south leg is d cos of the angle from north and the east or west leg is d sin of that angle. Use tan to find an unknown bearing from the two legs.
Which way is a bearing measured?
Always clockwise from north. North is 000 degrees, east is 090 degrees, south is 180 degrees and west is 270 degrees, and a full turn brings you back to 360 degrees, which is north again.