Year 11 Standard · Unit 2 Right-angled triangles and circle geometry · Lesson 2 of 8

Bearings: measuring a direction from north

A ship leaves port and the keeper at a lighthouse watches it go. This lesson turns a bearing and a distance into east and north, and back again.

Geometry and TrigonometryGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Year 7: an angle is 135°. How much more is needed to make a full turn of 360°?
    Answer225°, because 360 − 135 = 225.
  2. From Year 9: a right-angled triangle has sides 9 and 12 around the right angle. Find the hypotenuse.
    Answer15, because 81 + 144 = 225 and √225 = 15.
  3. From Year 11: from a point 20 m from a wall, the angle of elevation of the top is 40°. Which ratio finds the height?
    AnswerTangent: the 20 m is adjacent and the height is opposite, so height = 20 × tan 40°.
Words used in this lesson
bearing
a direction given as the clockwise angle from north. For example: a compass reading of a ship from a port.
three-figure bearing
a bearing written with three digits. For example: a bearing of 5° written 005°.
back bearing
the bearing of the start as seen from the finish. For example: a lighthouse seen from the boat.
clockwise
turning the way clock hands turn. For example: from north towards east.
due
exactly in the direction named. For example: due south.
component
one part of a movement, along one direction. For example: the east part of a journey.

The ship and the lighthouse

A keeper at a lighthouse sights a ship on a bearing of 070°. The captain of the ship radios back and wants to give the bearing of the lighthouse as seen from the ship.

Predict first: should the captain say 070° as well, or a different bearing? Write down what you think and the reason, then test it with the first figure.

110
12
290°back bearing
11.3 east, −4.1 northeast and north parts

Bearing 110°, 12 km: back bearing 290°. East 12 × sin 110° ≈ 11.3, north 12 × cos 110° ≈ −4.1 (a negative north is south).

Show me the steps
  1. Bearing 110°, 12 km: back bearing 290°. East 12 × sin 110° ≈ 11.3; north 12 × cos 110° ≈ −4.1, so the ship has gone a little south.
  2. Make the bearing 030° and keep 12 km: back bearing 210°. East ≈ 6.0 and north ≈ 10.4, both positive: north-east.
  3. Make the bearing 250°: back bearing 070°. East ≈ −11.3 and north ≈ −4.1, both negative: south-west.
  4. Make the bearing 000°, due north: back bearing 180°, east 0.0 and north 12.0. Every bearing and its back bearing differ by half a turn.

A bearing is the angle from north, measured clockwise, written with three figures. The back bearing differs from the bearing by exactly 180°. A ship that sails d on bearing b has gone d × sin b to the east and d × cos b to the north; a negative value means west or south.

Going the other way: from east and north, find the acute angle in the right-angled triangle with tan−1(east ÷ north), then turn it into a bearing by the quadrant: north-east is the angle, south-east is 180° − the angle, south-west is 180° + the angle, north-west is 360° − the angle.

4
−3
126.9°bearing from the origin
5.0distance from the origin

East 4, north −3: the point is south-east. Acute angle 53.1° from south, so the bearing is 180° − 53.1° = 126.9°, at distance 5.0.

Show me the steps
  1. East 4, north −3: south-east. The acute angle from south is tan⁻¹(4 ÷ 3) ≈ 53.1°, so the bearing is 180° − 53.1° = 126.9°. The distance is 5.0.
  2. East 4, north 3: north-east. The acute angle from north is 53.1°, and the bearing is just 053.1°.
  3. East −4, north −3: south-west. The acute angle from south is 53.1°, so the bearing is 180° + 53.1° = 233.1°.
  4. East −4, north 3: north-west. The bearing is 360° − 53.1° = 306.9°. The same triangle gives four bearings, one in each quadrant.

Count from north, and go clockwise. A ship going south-west has a bearing near 225°, not 45°. And a bearing of 5° is written 005°, with three figures, so that a bearing read aloud or written quickly cannot be misread.

Worked examples

1. A ship sails 25 km on a bearing of 038°. How far east and how far north does it go, to 1 decimal place?

  1. The 25 km is the hypotenuse. The east part is opposite the 38° angle at the start: 25 × sin 38° = 15.4 km.
  2. The north part is adjacent: 25 × cos 38° = 19.7 km.

2. Hikers walk 6 km east and then 4 km south. What is the bearing of the finish from the start, to the nearest degree?

  1. The acute angle from south is tan−1(6 ÷ 4) = 56.3°.
  2. South-east: bearing = 180° − 56.3° = 123.7°, which is 124°.

3. The bearing of a village from a farm is 164°. What is the bearing of the farm from the village?

  1. The back bearing is 180° different: 164° + 180° = 344°, which is less than 360°.
  2. So the bearing of the farm from the village is 344°.

Your go. The first step is given. A plane flies 90 km on a bearing of 213°. How far west of its start is it, to 1 decimal place?

  1. 213° is 33° past due south, so the acute angle in the triangle is 33° from south.
  2. The west part is opposite that angle. Which ratio, and what is 90 × that ratio?
Check the answer

West = 90 × sin 33° = 49.0 km. The south part is 90 × cos 33° = 75.5 km. Using the 213° straight in a calculator gives 90 × sin 213° = −49.0, and the minus sign just means west.

With a GDC (optional)

On many calculators sin and cos of an angle bigger than 90° give a signed answer, and the sign tells you the direction: a negative east value is west and a negative north value is south. That is a useful check, but the page also shows the method without it: take the acute angle in the triangle and work out the sides, then say the direction in words.

The inverse tangent key returns an acute angle for a positive ratio. Add it to, or take it from, 0°, 180° or 360° according to the quadrant, as on the page. Everything here can be done with a scientific calculator.

Check before you go on

1. The bearing of B from A is 305°. What is the bearing of A from B?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A ship sails 40 km on a bearing of 065°. How far east of its start is it, to 1 decimal place?

3. The bearing of town P from town Q is 238°. What is the bearing of Q from P, as a number of degrees?

Level 2 · Building

4. A hiker walks 9 km due east and then 5 km due north. What is the bearing of her finishing point from the start, to the nearest degree?

5. A ship sails 24 km on a bearing of 148°. How far south of its start is it, to 1 decimal place?

6. A plane flies 120 km on a bearing of 215°. How far west of its start is it, to 1 decimal place?

Level 3 · Stretch

7. A boat sails 10 km on a bearing of 040°, then 6 km on a bearing of 130°. How far is it from its start, to 1 decimal place?

8. A walker goes 8 km on a bearing of 110° and then 5 km on a bearing of 020°. How far north of the start is she, to 1 decimal place?

Something to think about

A ship sails 10 km on a bearing of 090°, then 10 km on a bearing of 180°. On what bearing is it from the start? Is it 135° exactly? The second figure stops at 8, so try east 8 and north −8, which is the same direction.

One way to think about it

Yes: it is 135°. East 10 and north −10 make an isosceles right triangle, so the acute angle from south is 45° and the bearing is 180° − 45° = 135°. The distance is 10√2 ≈ 14.1 km, not 20.

Exit ticket

9. A drone flies 50 m on a bearing of 303°. How far west of its start is it?