Hong Kong
Stage 1 - Stage 3

# Applications Including Bearings

Lesson

## True Bearings

In surveying and air navigation a true bearing measures the angle that a path or line makes with a ﬁxed north-south line. A true bearing measures clockwise from North ALWAYS. For example in a true bearing we would measure this obtuse angle and say it has a bearing of $\theta$θ$^\circ$°T.

We also write true bearings using exactly three digits. So if $\theta$θ is less than $100$100 degrees, then the true bearing will look something like $040$040$^\circ$°T or $008$008$^\circ$°T.

## Compass Bearings

The four main directions of a compass are known as cardinal points. They are north (N), east (E), south (S) and west (W).

A compass bearing is measured using the acute angle from the north south line.

we would write the bearing of B from A as S$\theta$θ$^\circ$°E.

## Which one first?

The bearing needed or used completely depends on which position comes first. Have a look at the investigation below, it quickly shows you how the angle changes depending on if we are measuring the bearing of A from B or B from A.

#### Example

A slightly lost hiker walks $300$300 m east before turning south and walking another $800$800 m. What is its true bearing from their original position (to $1$1 decimal place)? And then what is this as a compass bearing?

Think: to do this questions a diagram is a great way to start. This is the diagram I drew.

The angle we want to find is in green. It is equal to $90$90$^\circ$°$+$+the angle inside the triangle, which we can find using trigonometry.

Do: Bearing is

 $90+\tan^{-1}\frac{800}{300}$90+tan−1800300​ $=$= $90+\tan^{-1}\frac{8}{3}$90+tan−183​ $=$= $90+69.44$90+69.44 $=$= $159.44$159.44

and to one decimal place the final bearing is $159.4$159.4$^\circ$°T.

As a compass bearing we need to know the acute angle with the North South line. This is $180-159.4=20.6$180159.4=20.6

The compass points we will use will be South first, and then East, so the compass bearing is S$20.6$20.6$^\circ$°E.

#### Practice questions

##### Question 1

Consider the point $A$A.

1. Find the true bearing of $A$A from $O$O.

2. What is the compass bearing of point $A$A from $O$O?