topic badge

8.05 Bearings

Lesson

In surveying and navigation, bearings are used to help identify the location of an object.

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

True bearings

Also known as a three-figure bearing, a true bearing is:

  • measured from north ($N$N)
  • measured in a clockwise direction
  • written using three figures

$T$T is often but not always used to indicate a true bearing. If the angle measure is less than $100^\circ$100° it would be written like $040^\circ$040° or $040^\circ T$040°T.

To use true bearings to describe the location of a plane at point $B$B from the airport at point $A$A :

  1. place the centre of a compass on the starting point, in this case the airport.
  2. starting at North, rotate clockwise until we get to the line $AB$AB.
  3. write the angle as the true bearing of point $B$B.

 

The bearing of $B$B from $A$A is $127^\circ$127° or $127^\circ$127°$T$T.

The diagram below describes the bearing of $P$P from $O$O. Rotating clockwise from North, we get an angle of $55^\circ$55°.

Since this measure is less than three digits, we put a $0$0 in front of it so the true bearing of $P$P is $055^\circ$055°.

Consider the bearing of $O$O from $P$P . Since alternate angles are equal and we are starting at $P$P the true bearing would be $180^\circ+55^\circ=235^\circ$180°+55°=235°.

 

Compass bearings

A compass bearing describes the location of a point using:

  • the starting direction of either north or south;
  • the acute angle needed to rotate
  • the direction to rotate, east or west.

To describe the position of point $B$B from $A$A we would say:

"Starting at South, I then rotate $53^\circ$53° towards East."

 

We can write this mathematically as $S\ 53^\circ E$S 53°E

Worked examples

Example 1

Find the true bearing and compass bearings of point $P$P from $O$O.

True bearing:

Starting at North rotate in a clockwise direction.

$360^\circ-47^\circ=313^\circ$360°47°=313°

The true bearing of $P$P from $O$O is $313^\circ T$313°T.

Compass Bearing

Point $P$P is closest to North, so starting at North, rotate $47^\circ$47° towards West.

The compass bearing of $P$P from $O$O is $N\ 47^\circ W$N 47°W.

 

Which one first?

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

 

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?

    $\editable{}$ $\editable{}$$^\circ$° $\editable{}$

Question 2

What is the true bearing of Southwest?

Question 3

In the figure below, point $B$B is due East of point $A$A. We want to find the position of point $A$A relative to point $C$C.

  1. Find the true bearing of point $A$A from point $C$C.

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

    $\editable{}$ $\editable{}$$^\circ$° $\editable{}$

 

Applications

Applications cover many fields in navigation and surveying including search and rescue, mining and tunnel engineering.

Worked example

Example 2


A slightly lost hiker walks $300$300 m east before turning south walking another $800$800 m. What is the hikers three-figure and compass bearings from the original position (rounded to one decimal place)?

Think: It will be easiest to first visualise the situation by drawing a diagram, such as this one:

For the three-figure bearing, the angle we want to find has been marked. It is equal to $90^\circ$90°$+$+the angle inside the triangle, which we can find using trigonometry. We can then use this to find the compass bearing.

Do: The three-figure bearing is:

$90+\tan^{-1}\left(\frac{800}{300}\right)$90+tan1(800300) $=$= $90+\tan^{-1}\left(\frac{8}{3}\right)$90+tan1(83)
  $=$= $90+69.44395\ldots$90+69.44395
  $=$= $159.4$159.4 (one d.p.)

So the three-figure bearing is $159.4$159.4$^\circ$°.

To determine this as a compass bearing, we want to know the acute angle made with the North-South line. This is $180^\circ-159.4^\circ=20.6^\circ$180°159.4°=20.6°.

Looking back at the diagram, we can see that this bearing is closer to South, and on the East side, so the compass bearing is $S\ 20.6^\circ E$S 20.6°E.

Practice questions

Question 4

The position of a ship S is given to be $20$20 kilometres from P, on a true bearing of $0$0$49$49$^\circ$°T.

The position of the ship can also be given by its $\left(x,y\right)$(x,y) coordinates.

  1. If the ship's $x$x-coordinate is $x$x, find $x$x to one decimal place.

  2. If the ship's y-coordinate is $y$y, find $y$y to one decimal place.

Question 5

A boat travels $S$S$14^\circ$14°$E$E for $12$12 km and then changes direction to $S$S$49^\circ$49°$E$E for another $16$16 km.

  1. Find $x$x, the distance of the boat from its starting point. Give your answer to two decimal places.

  2. Find the angle $b$b as labelled in the diagram. Express your answer to the nearest degree.

  3. Hence write down the bearing that the boat should travel on to return to the starting point.

    $\editable{}$$\editable{}$°$\editable{}$

Outcomes

2.1.1.4

solve two-dimensional practical problems involving the trigonometry of right-angled and non-rightangled triangles, including problems involving angles of elevation and depression and the use of true bearings

What is Mathspace

About Mathspace