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.
ExampleA 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.
NoteThe 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
Bearings
Bearings are used to describe directions and are commonly applied in navigation, surveying, and other fields requiring precise directional information.
Three-Figure Bearings
A three-figure bearing is measured clockwise from the North, always given as three digits from 000° to 359°.
ExampleTo express East as a bearing: 090° To express South-West as a bearing: 225°
Calculating Bearings
Bearings can be calculated using trigonometric functions, often requiring the use of non-right angled triangle formulas such as the sine and cosine rules.
ExampleIn a national park, there are three locations: the visitor center $V$, the campsite $C$, and the restroom $R$. The campsite is directly north of the visitor center. The restroom is at a bearing of $068°$ from the visitor center and $135°$ from the campsite. The distance between the campsite and the restroom is $5\ km$. Find the distance between the visitor center and the restroom.
Solution:
For these types of questions, it's always best to draw a diagram:
Using this, we can work out that the angle $\angle VCR$ is $45°$. From here, we can use the sine rule to find $x$.
$\frac{x}{\sin45°} = \frac{5}{\sin68°}$
$x = \frac{5\sin45º}{\sin68º} \approx 3.81\ km$
TipFor application-based trigonometry questions, a very good skill to have is the ability to translate a word problem into a diagram. Reference the diagrams we included to help develop your own intuition for visualizing these problems.
