Practice Maths

L17 — Bearings

Key Terms

True bearing
A 3-digit angle measured clockwise from North, from 000° to 360° (e.g. 045°, 270°).
Compass bearing
Stated as a starting direction (N or S), then an angle toward E or W (e.g. N45°E, S30°W).
Back-bearing
The reverse direction of a bearing. Add 180° if the bearing is less than 180°; subtract 180° if it is 180° or more.
North component
How far north a journey travels: d × cos θ (negative result = south).
East component
How far east a journey travels: d × sin θ (negative result = west).
Displacement
Net position change from start to finish, found by adding north and east components of each leg.

Two Notation Systems

SystemDescriptionExample
True bearing3-digit angle, clockwise from North065°, 180°, 315°
Compass bearingStarting direction (N or S) then angle toward E or WN65°E, S45°W
  • Always measure bearings clockwise from North.
  • True bearings always use three digits: e.g., 045°, not 45°.
  • The back-bearing is the reverse direction: add or subtract 180°.
  • To find north/east components of a journey on bearing θ: North = d×cosθ, East = d×sinθ.
A N B 060° E S W Back-bearing: 060° + 180° = 240° d km
True bearing of B from A is 060°; back-bearing of A from B is 240°.
Hot Tip: True bearings always use three digits — write 045°, not 45°. For back-bearings: if the forward bearing is less than 180°, add 180°; if it is 180° or more, subtract 180°.

Reading and Writing Bearings

A bearing of 060° means “turn 60° clockwise from North.” Compass form: N60°E.

Key conversions:

TrueCompassDirection
000°NDue North
090°EDue East
135°S45°ESouth-East
270°WDue West
315°N45°WNorth-West

North/East Components

For a journey of distance d on bearing θ:

North component = d × cos θ    East component = d × sin θ

Negative north = south; negative east = west.

Worked Example — Navigation

A ship sails 20 km on bearing 130°. How far south and how far east has it travelled?

North = 20×cos130° = 20×(−0.6428) = −12.86 km ⇒ 12.86 km south

East = 20×sin130° = 20×0.7660 = 15.32 km east

  1. State the true bearing. Fluency

    • (a) Due North.
    • (b) Due East.
    • (c) South-West (45° into the SW quadrant).
    • (d) South-South-East (halfway between South and South-East).

  2. Convert between true bearing and compass bearing. Fluency

    • (a) True bearing 045° → compass bearing.
    • (b) Compass bearing S60°W → true bearing.
    • (c) Compass bearing N30°W → true bearing.
    • (d) True bearing 110° → compass bearing.

  3. Back-bearings. Fluency

    • (a) Bearing from A to B is 060°. What is the bearing from B to A?
    • (b) Bearing from A to B is 310°. What is the bearing from B to A?
    • (c) Bearing N40°E. Find the back-bearing in both compass and true form.
    • (d) If Y is on bearing 155° from X, what is the bearing of X from Y?

  4. North and east components. Fluency

    For each journey, find how far north/south and east/west the traveller moves.

    • (a) 10 km on bearing 060°.
    • (b) 15 km on bearing 135°.
    • (c) 8 km on bearing 210°.
    • (d) 20 km on bearing 300°.

  5. Read from the diagram: navigation. Understanding

    The diagram shows point A with North marked. A journey of 10 km is made on bearing 060° to reach point B.

    N A B 060° 10 km E S
    • (a) Express the bearing 060° as a compass bearing (N__°E format).
    • (b) Find the north component of the journey (how far north is B from A).
    • (c) Find the east component of the journey (how far east is B from A).
    • (d) Find the back-bearing from B to A.

  6. Two-leg journey. Understanding

    A boat leaves port P, sails 6 km due East (090°) to point A, then 8 km due North (000°) to point B.

    • (a) How far north is B from P?
    • (b) How far east is B from P?
    • (c) Find the straight-line distance from P to B.
    • (d) Find the bearing from P to B (to the nearest degree).

  7. Find the bearing between two points. Understanding

    Point A is 5 km north and 3 km east of point B.

    • (a) Draw a diagram showing A and B with a North arrow at B.
    • (b) Find the bearing of A from B (to the nearest degree).
    • (c) Find the bearing of B from A.
    • (d) Find the straight-line distance from B to A.

  8. Two ships from port. Understanding

    Two ships leave port P at the same time. Ship 1 sails 30 km on bearing 070°. Ship 2 sails 40 km on bearing 160°.

    • (a) Show that the angle between the two ships’ paths (at P) is 90°.
    • (b) Find the straight-line distance between the two ships.
    • (c) Find the angle between 070° and the line joining the two ships (use tan).
    • (d) Hence find the bearing of Ship 2 from Ship 1.

  9. Return journey. Problem Solving

    A boat leaves harbour H and sails 15 km due North to point A, then 9 km due East to point B.

    • (a) Find the straight-line distance from H to B.
    • (b) Find the bearing from H to B (to the nearest degree).
    • (c) Find the bearing from B back to H.
    • (d) How far is the total journey (H → A → B) compared with the direct route H → B?

  10. Search and rescue. Problem Solving

    A rescue helicopter leaves base B. It flies 80 km on bearing 025° to search area A. It then flies 60 km on bearing 115° to search area C.

    • (a) Find the north and east components of the B→A leg.
    • (b) Find the north and east components of the A→C leg.
    • (c) Find the total displacement from B: how far north and how far east is C from B?
    • (d) Find the straight-line distance from B to C and the bearing from C back to B.
See Answers ➔