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Applications Including Bearings



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.


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+tan1800300 $=$= $90+\tan^{-1}\frac{8}{3}$90+tan183
  $=$= $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?

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

Question 2

What is the true bearing of Southwest?

Question 3

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 4

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$° $\editable{}$

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