At the end of the lesson, students should be able to;

- Demonstrate a clear understanding of bearings.
- Calculate the magnitude and the direction of any point.

There are many situations in which you might need to describe your position and direction of travel. For example, you may tell a friend that you will meet them by the bus station, at 7pm, and that if they are late you will start walking up the high street. In mathematics, we use more precise ways to describe position and direction of travel.

Angles have many applications in our everyday lives. Builder architects for example use angles in their work. In geography angles are used to show directions. Sailors have no road to follow just a compass and a map to navigate with.

Surveyors also use angles and bearing in their work. Bearing gives direction in terms of an angle.

The above compass shows degree measurements from 0° to 360° in 10° intervals with:

- North representing 0° or 360°
- East representing 90°
- South representing 180°
- West representing 270°

The directions half way between the cardinal points are marked in the figure below.

We state North (N) or South(S) first and then the angle we turn through towards the East (E) or the West (W)

To state the direction of a point, write:

- N or S which is determined by the angle being measured.
- The angle between the North or South line and the point, measured in degrees.
- E or W which is determined by the location of the point relative to the North-South line

E.g. In the above diagram, the direction of:

*A*from*O*is N30^{0}E.*B*from*O*is N60^{0}W.*C*from*O*is S70^{0}E.*D*from*O*is S80^{0}W.

__Example____ 1__

Represent N50^{0}E and S40^{0}W on different cardinals.

__Solution __

N50^{0}E means from N measure 50^{0} towards E. S40^{0}W means from S measure 40^{0 }towards W.

__True Bearing__

The **true bearing** to a point is the angle measured in degrees in a clockwise direction from the north line. We will refer to the true bearing simply as the **bearing**.

A three figure bearing is always;

**An angle measured in degrees.****Measured as angles in the clockwise direction from the geographic North.****An angle given in three figure or digit from 000**^{0}to 360^{0}.

__Note __

You use extra zeros to make the number up to three digits if you need to.

For example, 3^{0} gives a bearing of 003^{0}and 99^{0} gives a bearing of 099^{0}.

__Example 2__

Look at the diagram below:

State the bearing of P and the bearing of Q.

__Solution __

The bearing of point ** P** is 065

The bearing of point ** Q** is 300

__Example 3__

State the bearing of the point *P* in each of the following diagrams:

__Solution__

a) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point ** P**.

The bearing of point ** P **is 048°.

b) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point ** P**.

The cardinal point S corresponds to 180°. It is clear from the diagram that the required angle is 60° larger than 180°. So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point ** P** is 180° + 60° = 240°.

So, the bearing of point ** P **is 240°.

c) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point *P*.

The cardinal point S corresponds to 180°. It is clear from the diagram that the required angle is 40° less than 180°. So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point *P* is 180°*– *40° = 140°.

So, the bearing of point *P* is 140°.

d) Mark the angle in a clockwise direction by indicating the turn between the north line and the line joining the centre of the compass to the point ** P**.

The cardinal point W corresponds to 270°. It is clear from the diagram that the required angle is 20° larger than 270°. So, the angle measured in a clockwise direction from the north line to the line joining the centre of the compass to point ** P **is 270° + 20° = 290°.

So, the bearing of point ** P** is 290°.

__Example 4__

Points of the compass can all be converted into bearings.

Find the bearings for:

(a) East (E)

(b) South (S)

(c) South-East (SE)

__Solution__

**REMEMBER: Bearings are always measured in a clockwise direction from the North and are given as 3 digits.**

__Example 5__

Describe each of the following bearings as directions.

a) 076°

b) 150°

c) 225°

d) 290°

__Solution__

a) The position of a point *P*on a bearing of 076° is shown in the following diagram.

The position of the point *P* is 76° east of north. So, the direction is N76°E.

b) The position of a point *P* on a bearing of 150° is shown in the following diagram

The position of the point *P* is 180° *–* 150° = 30° east of south. So, the direction is S30°E.

c) The position of a point *P* on a bearing of 225° is shown in the following diagram.

The position of the point *P* is 225° *–* 180° = 45° west of south. So, the direction is S45°W.

d) The position of a point *P* on a bearing of 290° is shown in the following diagram.

The position of the point *P* is 360° *–* 290° = 70° west of north. So, the direction is N70°W.

__Example 6__

Give:

(i) the compass bearing and the

(ii) true bearing of A from O for each of the following.

__Solution __

a) (i) compass bearing is N80^{0}W

(ii)True bearing of A from O = 360^{0}– 80^{0}

= 280^{0}

b) (i) compass bearing =180^{0}– 160^{0}

^{ }= 20^{0}

Compass bearing = S20^{0}E

(ii) True bearing of A from O = 160^{0}T

c) (i) OA is half-way between south and west.

Compass bearing is S45^{0}W

180^{0}+ 45^{0} = 225^{0}

(ii) True bearing of A from O = 225^{0}

__Unit 10 Lesson 16: Exercise __

What angle do you turn through if you turn clockwise from:

- N to S
- E to W
- N to NE
- S to NE
- W to NW

6) For each of the following match the compass bearing of A from O below with the diagram supplied

**N80 ^{0}W S15^{0}W NW due north SE S30^{0}W**

7) Match each of the diagrams in question (6) to the corresponding true bearing.

- a) 280
^{0 }T b) 315^{0 }T c) 135^{0 }T^{ }d) 000^{0}T^{ }e) 210^{0 }T^{ }f) 195^{0}T

8) Give

(i) the compass bearing

(ii) true bearing of following

9) Draw a diagram to show each of the following compass bearings:

a) S30^{0}E b) S40^{0}W c) N30^{0}E d) N56^{0}W e) N20^{0}E f) S67^{0}E

10) Draw a diagram to show each of the following true bearings.

a) 300^{0}T b) 340^{0}T c) 304^{0}T d) 170^{0}T e) 145^{0}T f) 014^{0}T

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