Trigonometry

Lesson

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.

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.

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+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$180−159.4=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.

Consider the point $A$`A`.

Find the true bearing of $A$

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

`A`from $O$`O`?$\editable{}$ $\editable{}$$^\circ$° $\editable{}$

What is the true bearing of Southwest?

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.

If the ship's $x$

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

`y`, find $y$`y`to one decimal place.

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`.

Find the true bearing of point $A$

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

`A`from point $C$`C`?$\editable{}$ $\editable{}$$^\circ$° $\editable{}$