Trigonometry Research Paper - Measuring the Height of Buildings

It was necessary to undertake a look at the right angle trigonometry applications after having learned about it in class. Trigonometry is a mathematical branch that is concerned with studying the relationships involving the lengths and the angle of triangles. Originally, the field emerged from the Hellenistic world where geometry was applied in studying astronomies (Cohen, Lee & Sklar, 2006). During these studies, it was realized that the length of right angle triangles, as well as the angles existing between the sides, exhibited a fixed relationship. This implies that when the length of one side and at least one side of the triangle was known, it was possible to determine the length and angles of all others using algorithms. Later, these calculations were later known as trigonometric functions. The height of a building is often determined every time once construction is made or necessary investigation is required.

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Solving the height of a building is a real life case. The knowledge of right angle trigonometry can be applied in such a situations (Cohen, Lee & Sklar, 2006). Height for buildings is required under different circumstances which may include determining the level of access into the building, need to install water tanks or other finishing materials. The height of a building is characterized by right angle triangle. While studying geometry, it is highly likely that right angle triangles have been used when solving problems that involve distance by the use of Pythagorean Theorem. By incorporating the aspects of elevation and depression, bearing in mind that that the horizontal baseline to the building and the height of the building form a 90-degree angle and can refer to as legs of the right angle (McKeague & Turner, 2017). Figure 1 below shows the formation of a right triangle. The legs are described as either adjacent or opposite side to angle A.

Figure 1: Sketch of the Building

Figure 1 above is a sketch of the building being determined. The line intersecting the horizontal ground and the wall of the building forms a right angle triangle. Point A is the position of the observer and the hypotenuse represents the elevation line of sight of the observer. The intersection of the hypotenuse and the adjacent line forms the angle of depression (McKeague & Turner, 2017). For the purpose of this study, the right side will be referred to as opp, the hypotenuse side as hyp and adjacent side as Adj. As exhibited in the definitions below, sine is referred to as sin, tangent tan and cosine as cos. These terms originality is related to the tangents and arcs to circles.

Sin (A) = opp/hyp

Tan (A) = opp/hyp = sin (A) /cos (A)

Cos (A) = Adj/hyp

Besides, there are also special angles that need consideration. These are displayed in the following table

Table 1: Special Angles

Cos sin Tan

30 degree 3/2 1/2 3/3

45 degree 2/2 2/2 1

60 degree 1/2 3/2 3

Table 1 above shows special angles and their respective trigonometric relationships. Such knowledge is critical when understanding the applications of trigonometry knowledge of right angled triangles. The definitions revealed under table 1 are the special case of angles that can be used when applying the trigonometric knowledge in calculating the height of buildings.

Solving the Height of the Building

In this section, we examine different approaches that can be used when solving the height of the building. Here, Pythagorean Theorem knowledge and six trigonometric functions are used. While understanding that a right triangle is the one with a 90-degree angle, solving it requires one to find the measurements of one or both angles (Cohen, Lee & Sklar, 2006). The approach to finding the solution depends on the amount of information available. At the same time, the tangent function can be utilized to find the solution since the measurements of the opposite side is available. In this case, it is evident that there are different ways of finding missing sides, however, this should not be a confusion rather a display of ability to explore a variety of sides which should be employed when finding the solution (Cohen, Lee & Sklar, 2006). Besides, when identifying one side of a triangle, one can use trig ratio or Pythagorean Theorem. When solving the height, in this case, it is important to identify measures of all the angles.

Furthermore, the length of two sides should be identified. The lengths can be found by the use of the trig ratio. Once this is established, it is then possible to find the length of the third side by using Pythagorean Theorem or the trig ratio (Cohen, Lee & Sklar, 2006).

Angles of Depression and Elevation

To complete the solution, there is need to establish the angle of depression and elevation depending on the position of the observer. The concept of right angle triangle is highly required to complete this part. Figure 3 below shows these angles.

Figure 4: Depression and Elevation Angles

(Source: McKeague & Turner, 2017)

An elevation angle is the one between the horizontal line of sight and the line of sight above the object (Zill & Dewar, 2012). The depression and the elevation angle scales how the observer is looking at the top and bottom of the building. The object above the observer represents the top of the building while the object below the observer is the base of the building. For instance, it is possible to measure an angle of elevation when one is on the ground and looking up on the top of a tree. On the other hand, the angle of depression represents the angle between the horizontal line of sight and the line of sight down to the object, for example, when looking at the foot of a tree (Zill & Dewar, 2012). Once all these aspects are established, it is now possible to calculate the height of the building. Assuming that the distance where the observer looking at the top of the building is standing at a distance of 20 meters from the wall of the building, while the angle of elevation identified is 42 degrees, then the height of the building is established as follows;

Figure 4: Depression and Elevation Angles

Based on the information given under the figure above, it is possible to find the size of side H. To find the total length of the building, an addition of extra 6 meters is required (Zill & Dewar, 2012). Finding H requires the use of tangent values as shown below;

Tan (42 degrees) = opp/ adj = H/ 20

Thus, through close multiplication, H = 20 tan (42 degrees) 45.83. Therefore, the height of the building is 45.83 + 6 51.83 meters.

Conclusion

The study has examined the calculation of the height of a building as a real application of right angle triangle trigonometry when solving actual world problems that involve the right angle (Cohen, Lee & Sklar, 2006). When finding the distances or lengths, the angles of elevation, the angles from the bearings in the navigation, the depression angles and other actual situations which results to right triangles. As exhibited in the paper, trigonometric knowledge is vital as it makes it easy to find different strategies when solving right angled problems (Zill & Dewar, 2012). The study has attempted to explore the application of the knowledge in finding the solution to both non right and right angle triangles as well as the angles of depression and elevation as the fundamental areas. From the exhibition and analysis of the outcome of the study, the information that should be noted is that when finding strategies of applying the knowledge, one has to understand the type of situations where right triangles emerge and whether there are cases where it is not possible to solve the right triangles. However, it should be understood that trigonometry can be applied in solving both earthly ad astronomical problems of the wide variety. Hence, it is an important component not only in mathematics but also in other fields.

References

Cohen, D., Lee, T. & Sklar, D. (2006). Precalculus with unit circle trigonometry. Belmont, CA: Thomson-Brooks/Cole.

McKeague, C. & Turner, M. (2017). Trigonometry. Boston, MA: Cengage Learning.

Zill, D. & Dewar, J. (2012). Algebra and trigonometry. Sudbury, MA: Jones & Bartlett Learning.