Introduction
Heights and Distances
Horizontal Level and Line of Sight
Angle of Depression
Angle of Elevation
Calculating Heights and Distances
Frequently Asked Questions
Conclusion
Trigonometry is a branch of math that studies the relation between the angles and the lengths of the sides of the triangles. There are various ways in which people can apply trigonometry in real-life situations. Trigonometry is especially important to subjects like geography and astronomy. For example, it is an essential tool to construct maps. It also helps in locating continents and islands based on the latitudes and longitudes. This chapter explores some applications of trigonometry in detail.
One of the primary applications of trigonometry is to find the distance between two or more places or to find the height of objects without measuring the height or angle of that object. Therefore, trigonometry proves to be beneficial to cartographers, navigators, sailors, astronomers, and architects who use it to determine the heights of objects and distances between two or more things. There are two types of measurements for objects: Angle of depression and angle of elevation.
Before knowing in detail about the types of measurements, it is essential to understand the horizontal level and the line of sight. The horizontal level is the line horizontal to the eye level of the observer. For example, suppose the observer is looking from point E to point Q, which is horizontal to point E, then the horizontal line becomes the line between E and Q, which is EQ. Therefore, the horizontal line is directly related to the point from where the observer is looking to the point where they are looking.
On the other hand, the Line of Sight is the line between the observer's point of view to the object the observer is viewing. For example, suppose an observer is looking from point E towards object O. So, the Line of Sight will be EO. The line of sight is always within the perception range of the observer. The items they are looking at become the endpoint of the line of sight, and their eye or the place where they are viewing the object from becomes the starting point of the line of sight.
The angle of depression is the angle present in the objects below the horizontal level, i.e., it is relevant for the things below the flat eye level of the observer. The horizontal level forms the angle of depression to the line of sight. When the eye or the place from where the observer is looking is above the object, they perceive then the angle of depression forms. In cases where this angle occurs, the observer has to look below the eye level, thus deviating from the horizontal eye level to look at the object they are viewing.
For example, suppose an observer looks at a fruit fallen from the tree's ground. Since the eye of the observer, which is on the tree, is above the fruit, which is on the floor, and below the tree, the observer has to lower their perception level to look at the fruit. Consequently, this deviates the horizontal line, which is supposed to be flat to the eye level of the observer to the ground. If the point from where the observer is viewing is Point Q and the fruit which the observer is viewing is Point P, then the horizontal line from which the observer deviates is R, the angle of depression or angle θ forms between Point R, Point Q, and Point P.
The angle of elevation, like the Angle of Deviation, is another type of measurement of distances. It is present in the objects above the horizontal level, i.e., it is relevant for the things above the eye level of the observer. The angle is formed by the line of sight and the horizontal level. When the eye or the place from where the observer is looking is below the object level, their perception of the streetlight elevates their eye level. The line of sight forms the angle of elevation from the horizontal level of the eye; in cases where the angle of elevation forms, the observer has to look above eye level, thus deviating from the original horizontal eye level of the observer.
For example, suppose an observer is looking up at the street light from the road. The observer immediately has to look up to perceive the street light, which makes them deviate from the horizontal level of the observer. If the eye or the point from where the observer is looking is point P and the streetlight is Point Q, and the original horizontal level is Point E, then the angle of elevation angle θ forms between Points E, Q, and P.
Trigonometric Ratios are essential to the calculation of heights and distances of objects. Firstly, it is vital to draw the diagram corresponding to the problem. Without a visual representation of the problem, it won't be easy to calculate the heights and distances of objects. After drawing the diagram, the next step is to mark all the measurements of the things and mark the distances. The last step is to use the trigonometric ratios to find out the objects' unknown lengths, distances, and angles. This way, it will be easier to calculate the heights and distances using trigonometry.
Why is it important for cartographers to learn trigonometry?
Cartographers are people who make maps. Maps require locating different continents and islands to plot them on paper eventually. Trigonometry helps cartographers find islands and continents and place them correctly on paper with the lep of latitudes and longitudes. Trigonometry also helps the cartographers measure the continents and islands' heights, sizes, and lengths to create maps. Therefore, cartographers need to learn trigonometry to create appropriate maps and plot the locations of the continents and islands correctly.
What is the angle of elevation?
The angle of elevation is formed by the line of sight and the horizontal line. When the observer looks at an object above the point of their eye level, they have to look up to view it automatically. Consequently, this results in the deviation from the horizontal eye level. The horizontal line eventually shifts to connect with the line of sight between the observer and the object, which forms the angle of elevation. Therefore, the angle of elevation is formed above the eye level of the observer.
What is the Angle of Depression?
The angle of depression is also formed by the line of sight and the horizontal level. When the observer looks at an object below the point of their eye level or where they are standing, they have to look down to view it automatically. Consequently, this results in the deviation from the horizontal eye level. The horizontal line eventually shifts to connect with the lien of sight between the observer and the object, which forms the angle of deviation. Therefore, the angle of depression is formed below the eye level of the observer.
How to measure celestial bodies with trigonometry?
It is essential to use the parallax method to measure celestial bodies with trigonometry. The parallax method measures long distances between the observer and the objects. The parallax method produces the parallax angle that is half the angle between two lines of sight when the observer views the things from two different places. Subsequently, people, especially astronomers, can measure the long distances between the celestial bodies with the parallax angle and the distance between the objects. Therefore, astronomers can measure the celestial bodies and the long distances between them with the knowledge of trigonometry.
What is the line of sight and horizontal lines?
Line of sight is the line created between the observer and the object they are viewing. If the eye or the place from where the observer is seeing is E and the thing they are looking at is Q, then the line of sight is EQ which is between E and Q. The horizontal lines are the lines formed between the eye or the place from where the observer is looking and the horizontal line through the eye level of the observer. If the eye or the location from where the observer is looking is E and the horizontal point is P, then the horizontal one is EP.
Trigonometry is essential for numerous things and countless real-life situations. Therefore, students must understand this chapter with utmost focus. Furthermore, they must use all the study materials and guidance required to understand this chapter.
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