A bearing describes direction as an angle measured clockwise from north, always given as three digits (like 045° or 270°). Right-angled trigonometry (sine, cosine and tangent) can be used to solve navigation problems involving bearings — like finding the distance and direction of the final leg of a journey when you know the other legs. Setting up these problems usually involves sketching a diagram with north lines at each point, marking the given angles and distances, and identifying which trigonometric ratio (or the sine/cosine rule for non-right-angled triangles) applies to find the missing information.
Example
A hiker walks 5km on a bearing of 060° from their starting point, then wants to know the bearing and distance needed to walk directly back to the start — sketching the triangle formed by the outward journey and the direct return path, with north lines marked at both the start and the current position, sets up the problem so the correct angle and trigonometric ratio can be identified.
Key terms
Bearing:
A direction expressed as an angle measured clockwise from north, given as three digits.
True north:
The reference direction (000°) from which bearings are measured.
Questions
1. A bearing describes direction as an angle measured:
Clockwise from north
Anticlockwise from south
From the nearest landmark only
Randomly, with no fixed reference
2. Bearings are always given as:
Three digits
A single digit
A word, like "northeast"
A fraction
3. A bearing of 090° points:
Directly east
Directly north
Directly south
Directly west
4. A bearing of 000° points:
Directly north
Directly south
Directly east
Directly west
5. A bearing of 180° points:
Directly south
Directly north
Directly east
Directly west
6. Solving navigation problems with bearings often uses:
Right-angled trigonometry
Only addition and subtraction
No mathematics at all
Only basic counting
7. A useful first step in solving a bearing problem is:
Sketching a diagram with north lines marked
Skipping straight to a calculation with no diagram
Ignoring the given angles entirely
Assuming all bearings are the same
8. A ship sails on a bearing of 045° for 10km. Roughly how far east has it travelled? (Use sin 45° ≈ 0.71)
About 7.1 km
About 10 km
About 5 km
About 14 km
9. A bearing of 270° points in which compass direction?
West
East
North
South
10. If you are travelling on a bearing of 060°, the bearing to return directly the way you came is:
240°
060°
300°
180°
11. A hiker walks 3km north, then 4km east. Using Pythagoras, approximately how far are they from their starting point in a direct line?
5 km
7 km
3 km
4 km
12. Why is a bearing always written using exactly three digits (e.g. 045° rather than 45°)?
It's a standard convention that avoids ambiguity and keeps all bearings a consistent, clearly recognisable format
Three-digit bearings represent a completely different, unrelated measurement to two-digit ones
This is a purely arbitrary rule with no practical reasoning behind it
Bearings under 100° are mathematically invalid without the extra digits
13. Why does solving a bearing problem typically require drawing a new north line at each new position, rather than just one at the very start?
Bearings are always measured relative to north at the specific point you're standing, so each position needs its own reference line for the angles to be correctly measured
A single north line drawn only at the start is always sufficient for solving any bearing problem
North lines are purely decorative and provide no actual mathematical function in these diagrams
The position from which a bearing is measured has no bearing on how the problem should be set up
14. Why might co-interior (allied) angle relationships between parallel north lines be useful when working out the bearing of a return journey?
North lines at different points are parallel, so angle relationships between parallel lines can help relate the outward and return bearings correctly
Parallel north lines never actually create any useful angle relationships in bearing problems
The bearing of a return journey can never be mathematically related to the bearing of the outward journey
Angle relationships between parallel lines are irrelevant to solving bearing and navigation problems
15. A ship sails 8km on a bearing of 030°, then 6km on a bearing of 120°. Since these two bearings are 90° apart, what tool could find the direct distance back to the start?
The Pythagorean theorem, since a right angle is formed between the two legs
Only guessing, since no mathematical method applies here
Addition of the two distances only, with no angle consideration
This problem cannot be solved with any known method
16. Why might the sine rule (rather than basic right-angled trigonometry) be necessary for some bearing and navigation problems?
Not all triangles formed in navigation problems are right-angled, and the sine rule works for any triangle where enough sides and angles are known
The sine rule is only ever useful for right-angled triangles, identically to basic trigonometry
All bearing and navigation problems can always be solved using only right-angled trigonometry
The sine rule has no genuine application in real navigation or bearing problems
17. A plane flies 200km on a bearing of 070°. Approximately how far north has it travelled? (Use cos 70° ≈ 0.34)
About 68 km
About 200 km
About 34 km
About 188 km
18. Why might real navigation systems (like GPS) still rely on the same underlying trigonometric principles used in bearing problems, even though the calculations are automated?
The mathematics of angles, distances and direction remains the same; automation just performs the same calculations faster and more precisely than by hand
GPS systems use a completely unrelated method with no connection to trigonometry or bearings
Automated navigation systems never actually rely on any mathematical calculation
The underlying mathematics changes completely once a calculation is automated by a computer system
19. Why might a navigation problem involving three or more legs of a journey (rather than just two) require breaking the path into multiple triangles solved in sequence?
Complex multi-leg journeys can be decomposed into simpler triangle problems, solved one at a time using the results carried forward from each previous step
Multi-leg journeys can never actually be broken down into smaller, more manageable triangle problems
A single trigonometric calculation is always sufficient for any journey, regardless of how many legs it involves
The number of legs in a journey has no bearing on how the navigation problem should be approached mathematically
20. Why might an orienteering course designer need to verify bearing calculations very precisely, given that even a small angular error can lead to a large positional error over a long distance?
A small error in angle compounds over distance, so a competitor following a slightly incorrect bearing could end up significantly off course after travelling a long way
Small angular errors in a bearing calculation never actually have any meaningful effect on final position, regardless of distance travelled
The distance travelled along a bearing has no bearing on how much a small angular error affects final position
Orienteering course design never actually requires any precise trigonometric calculation
21. Understanding bearings and navigation trigonometry mainly helps you to:
Apply trigonometric methods to solve real-world direction and distance problems
Assume all navigation problems can be solved without any diagram or reference to north
Ignore the distinction between right-angled and non-right-angled triangles in these problems
Treat bearings as unrelated to standard trigonometric ratios
Answer key (parent copy)
1. Clockwise from north
2. Three digits
3. Directly east
4. Directly north
5. Directly south
6. Right-angled trigonometry
7. Sketching a diagram with north lines marked
8. About 7.1 km
9. West
10. 240°
11. 5 km
12. It's a standard convention that avoids ambiguity and keeps all bearings a consistent, clearly recognisable format
13. Bearings are always measured relative to north at the specific point you're standing, so each position needs its own reference line for the angles to be correctly measured
14. North lines at different points are parallel, so angle relationships between parallel lines can help relate the outward and return bearings correctly
15. The Pythagorean theorem, since a right angle is formed between the two legs
16. Not all triangles formed in navigation problems are right-angled, and the sine rule works for any triangle where enough sides and angles are known
17. About 68 km
18. The mathematics of angles, distances and direction remains the same; automation just performs the same calculations faster and more precisely than by hand
19. Complex multi-leg journeys can be decomposed into simpler triangle problems, solved one at a time using the results carried forward from each previous step
20. A small error in angle compounds over distance, so a competitor following a slightly incorrect bearing could end up significantly off course after travelling a long way
21. Apply trigonometric methods to solve real-world direction and distance problems