Trigonometry · Ages 14–15
Bearings
True bearings: from north, clockwise, three digits; back ± 180°.
How to do it
Method
True bearings: from north, clockwise, three digits; back ± 180°
S40°E = 140°; the way back 320°
Another way
Split into east and north
d sin θ, d cos θ
In full
Bearings come two ways. A true bearing is measured clockwise from north, written with three digits: east is 090°, south 180°, south-west 225°. A compass bearing goes from north or south towards east or west, under 90°: N30°E is 030°, S40°E is 180° − 40° = 140°, S20°W is 200°, N50°W is 310°. The way back (A from B) differs by 180°: B on 070° from A puts A on 250° from B. For a journey, draw it first with a north line at each point and look for a right angle: 3 km on 040° then 4 km on 130° meet at a right angle, so the end is 32 + 42 = 5 km from the start. Another way: split each leg into its east and north parts (d sin θ, d cos θ), add them, and use Pythagoras and tan⁻¹.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
B is on a true bearing of 305° from A. As a compass bearing, what is B's bearing from A?
- A. S55°W
- B. S55°E
- C. N55°E
- D. N55°W
Choose the right answer
Show the answer and working
Answer: D. N55°W
See which quarter 305° is in, then the angle from north or south, 55°.
Example 2
B is on a true bearing of 210° from A. What is A's true bearing from B, in degrees?
Show the answer and working
Answer: 30°
The way back differs by 180°: 210° ± 180° = 30° (kept between 0° and 360°).
Example 3
A is 34 km due north of O, and B 6 km due east of O. To the nearest 0.1°, what is B's true bearing from A?
Show the answer and working
Answer: 170°
Looking south from A, B is tan⁻¹(634) ≈ 10° towards east: S10°E of A, that is 180° − 10° ≈ 170° (rounded at the end).
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.B is S41°W of A. What is B's true bearing from A, in degrees?
Answer:
2.A ship sails 20 km from O on a bearing of 051° to A, then 21 km on 141° to B. How far is B from O, in km?
Answer:
3.B is on a true bearing of 073° from A. What is A's true bearing from B, in degrees?
Answer:
4.A is 35 km due north of O, and B 14 km due east of O. To the nearest 0.1°, what is B's true bearing from A?
Answer:
5.A plane flies 29 km from O on a true bearing of 151°. Correct to 1 decimal place, how far east has it moved, in km?
Answer:
All the answers
- 1. 221°
- 2. 29 km
- 3. 253°
- 4. 158.2°
- 5. 14.1 km
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
Start playingPlay and learn in the secondary world, Star Galaxy: animated methods, monster battles, a review of every mistake