Applications of Trigonometry

There are many applications of trigonometry, such as finding height of a building, aviation, architecture, engineering designs, etc.

Examples

1. The distance from where the building is observed is 90 metres from its base and the angle of elevation to the top of the building is 35$^\circ$. Find the height of the building.

Given the distance from the building is 90 metres and the angle of elevation from to the top of the building is 35$^\circ$.

$\tan { 35 } =\cfrac { height }{ 90 } \\ \\ height=90\tan { 35 } \\ \\ height=42.64\quad metres$

Three-dimensional trigonometry is widely used in modelling real-life applications.

2. From point X, 25 m due south of the base of a tower, the angle of elevation is 47$^\circ$. Point Y is 15m due east of the tower. Find:

a) the height, h, of the tower, correct to 1 decimal place

b) the angle of elevation, $\theta$, of the tower from point Y.

$from\quad \triangle XTO\\ \\ \tan { 47 } =\cfrac { h }{ 25 } \\ \\ h=25\tan { 47 } \\ \\ h=\quad 26.8\quad metres$
$from\quad \triangle YTO\\ \\ \tan { \theta } =\cfrac { 26.8 }{ 15 } \\ \\ \theta =\tan ^{ -1 }{ \cfrac { 26.8 }{ 15 } } \\ \\ \theta =60 ^\circ 46'$