Many problems involving angles, distances and directions can be solved using trigonometry.

All of the formulae of trigonometry from the cosine rule through to the double angle formulae are available to help solve these problems.

If the problem involves finding maximum and minimum values it may be necessary to differentiate.

Diagrams are also almost always of assistance when doing this type of problem.

Problems could be in two or three dimensions.

As every problem is different, there is no fixed way to approach them and in this topic, two examples will be given.

As always, practice makes perfect. Try to do as many of the questions in the exercise as possible.

Example 1


A chairlift rises in a straight line up a mountainside at an angle of 50°, as shown in the diagram.

When one chair has 200 m to travel to the peak, a second is at a vertical distance of 210 m below the peak.

Calculate the distance between the chairs.


Draw a labelled diagram.


The distance between the chairs is 74.1 metres(to 3 sig. figs.)


Example 2

Two towns Awanui and Bottomley, are on opposite sides of a hill with a cross section ASB, of height h metres.

The towns are connected by a straight tunnel AB, of length s metres, through the hill and a path ASB over the hill.

The path makes angles α and β with the tunnel at A and B respectively.

1. Show that the length of the tunnel is given by

s = Y12_Applications_of_Trigonometry_05.gif

2. Find the length of the path ASB over the hill if the tunnel is 1,500 m long and α = 20° and β = 30°.


1. Draw a diagram.




Adding x and (s − x):



2. The path ASB = AS + SB

First find h:


Y12_Applications_of_Trigonometry_13.gif Y12_Applications_of_Trigonometry_14.gif

The path ASB = AS + SB

= 980 + 670

= 1650 m

The length of the path over the hill is 1650 metres   (to 3 sig. figs.)