# 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.

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