Trigonometric Applications in Real-World Scenarios
Angles of Elevation and Depression
Ever wondered how people measured the heights of big things before the "measure" tool on phones? This is how. They had to look up tangents of angles in a book listing all of them, but luckily we have calculators.
Angle of Elevation
An angle of elevation is the angle formed between the horizontal line of sight and a line of sight pointing upwards towards an object. It is measured from the horizontal to the line of sight.
A person standing 20 meters away from a building looks up at the top of the building with an angle of elevation of 30°. To find the height of the building, we can use the tangent function:
$\tan(30°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\text{height}}{20}$
$\text{height} = 20 \tan(30°) \approx 11.55$ meters
Angle of Depression
An angle of depression is the angle formed between the horizontal line of sight and a line of sight pointing downwards towards an object. It is measured from the horizontal to the line of sight.
The angle of depression from point A to point B is equal to the angle of elevation from point B to point A.
ExampleA person standing on a cliff 100 meters high looks down at a boat in the sea with an angle of depression of 25°. To find the distance of the boat from the base of the cliff, we use the tangent function:
$\tan(25°) = \frac{\text{opposite}}{\text{adjacent}} = \frac{100}{\text{distance}}$
$\text{distance} = \frac{100}{\tan(25°)} \approx 213.79$ meters
