Research Paper on Applied Trigonometry and Geometry

Introduction

Achieving relevance and sense in the real world, mathematics enthusiasts, teachers, and scholars strive to relate mathematical concepts to real-life applications. Perhaps, these efforts make mathematics in general palatable. It is easy to learn and solve problems when one can refer to things that surround them. It is always intriguing to understand situations in an entirely new dimension. It is from such provocation of thought that we begin to explore situations further and to push ourselves to input our energies, time, and finances at times, to advance experience of specific phenomena. By extension, understanding our environment and being able to relate to it through mathematical concepts also allows us to explore new situations. However, many times mathematical concepts are passed on to learners in a way that leaves them wondering in what way they will ever apply mathematics in their life. Teaching and learning mathematics takes a format that probably does not provoke one observation of their physical world. Daily life, revolves around mathematics even though it may not involve calculations.

Looking up at a building or looking down from a sky-crapper for instance, involves distances and angles that are possible to translate into mathematical notation and calculate one or more of phenomena needed. Ability to do this solves not only one particular problem but also provides a view of the world that makes everything understandable and beautiful. This example is perhaps the most straightforward application of modern trigonometry. In the earliest use of these mathematical concepts, this branch of mathematics concerned with solving triangles was merely as a tool for astronomical calculation. Nevertheless, it went ahead to a subject of immense practical value to engineering, architecture, surveying, navigation, and astronomy. In more advanced applications, Trigonometry and geometry find a vital role in the analysis and representation of waves and other periodic phenomena in a visual manner (Nelson, 2008).

This research paper attempts to present a case for the real-life application of trigonometry in literally more playful situations such as in a basketball game. It will assess, whether it is possible to apply the concepts developed in the many years of development and application of geometry and trigonometry to score more baskets. While this assessment may seem far-fetched, given a precise observation, the game's translation into mathematical notation and evaluation of the distances and trajectory projection becomes a possibility. This paper touches briefly on the history of and the development of trigonometry in this introduction. The literature review, in turn, addresses already developed trigonometric quantities and their manipulation. The data collection section describes the experiment performed complete with the assumptions made and the illustrations of the setup. The results provide a formula for the addresses focuses on the use of geometry and trigonometry to find a specific trajectory and angle that would land a basketball in the basket.

Trigonometry Historical Development

Like the many traces of human civilization and writing from Egypt and Mesopotamia, the earliest traces of works around trigonometry were also found there. Historians have found stone tablets in Babylonian dating as early as 1900-1600 BC that listed ratios equivalent to the modern sec2. Additionally, the Egyptian Rhind papyrus dating around the same time, 1650 BC also contained similar ratios of the sides of a triangle applied to solving problems relating to pyramids. However, the Egyptians and the Babylonians did not have the present day reliance on the concept of angular measure and ratios. What the two civilizations had developed, therefore, could be regarded as properties of triangles as opposed to features of angles (Nelson, 2008).

Present in the trigonometry breast of change is the Greek who made further advances in the field from the time of Hippocrates of Chios (c.430 BC). Hippocrates studied the relationships between the arc of a circle and the chord of the arc. Others like Hipparchus produced a table of chords that could easily be seen as the precursor to the modern tables of sines. Menelaus of Alexandria worked to create the spherical trigonometry branch, while Ptolemy developed the chords of angles and investigated trigonometric identities. These works were a result of the mathematician tackling their problems of the time (Nelson, 2008). This version of Greek trigonometry developed further through works of scores of Hindu mathematicians who advanced to using half-chords of circles with given radii. Modern trigonometry traces its roots to the Renaissance mathematician Regiomontanus. The mathematician was the first to treat trigonometry as a disciple on its own. Through the works that followed, trigonometry rose to an important topic in mathematics for science and engineering.

Literature Review

Modern trigonometry uses the unit circle to extend the domain of sine and cosine to all real numbers. It is from the unit circle that process for determining the sine and cosine of any angle begins ("Trigonometry").

The above translates to the following that for any angle th it can be represented as follows:
It is through these trigonometric functions that the most fundamental mathematical application has been based. Given any of the two sides of a right-angled triangle, it is possible to calculate the angle between the two sides. Through the Pythagoras theorem, it is possible to find the missing side of a right-angled triangle and consequently the missing angle (Burger). A lot of problems in the introductory topics to trigonometry may revolve around revolve around solving for the required side or angle of a triangle (Rusczyk).

Data Collection

The Experiment

Setting up the experiment required that we use the standard measurements that apply to all levels of the game be it junior high school, High school, NBA or FIBA. The set up was as shown. The assumption is that the player uses the same amount of energy each time only varying the angle, a. The set up also assume that the player was of junior high school average height and the ball was of standard weight. Moreover, the set up too assumes there are two ways to score. A player can score directly into the 10 inches diameter ring. Alternatively, the player can score by hitting a target point above the ring on the backboard that deflects the ball into the ring. Therefore, the experiment did not concern itself with issues of projectile's acceleration and the applied force. Hence the investigation involved itself with collecting scalar quantities data to feed into the Pythagoras theorem represented by a2 + b2 = c2 , that is, the various side of a right-angled triangle (Brannan et al.).

For a player of average height, h, standing distance X from the basket, and determining a point in the path of the basketball to for the side of a right-angled triangle b, it was possible to collect data to calculate the angle through which players can successfully deliver a ball into the basket each time.

Data scoring directly in to the ring2512612328323

a=length from the player to the height gauge

b= height measured -h( height of the player)

This data was collected from various distances around the pitch, several times at each instant at different inclinations

Scoring through targeting a point above the ring

Similar information was collected for this scoring method maintaining same assumptions as in the first set up.

Results

As illustrated in the literature review, using trigonometry makes it possible to calculate the sine, cosine or tangent and solving for the required angle. From the curve traced by the basketball as it is thrown into the ring, the right-angled triangle is developed by drawing a tangent, of length c, to that curve. Solving for its length using the Pythagoras theorem

C= a2+b2Therefore, to find required angle the involves finding the sine and converting the figures arrived at into their angles (Zill, and Dewar). From the formulas shared above, then,

Sin th = a2+b2bTherefore the angle th = sin-1 ( a2+b2b )

While from experiment it is possible to determine many of the optimal angles required to get the ball into the basket, it is not possible to measure these angles while in the field of play during a tournament. However, through consistent practice trying to score from a various point on the pitch, it is possible inherently get this angle right consistently too.

Conclusion

Mathematics provides a solution to some daily life situations. These scenarios may not appear as a problem, but until we try to improve the way we experience them, it is then the appreciation of what role mathematics plays in everyday life. In solving the problems of the day, ancient times mathematicians continued the exploration of the properties of a triangle to improve on their space exploration and consequently achieve significant inroads into what is a modern subject, trigonometry. In this research paper to find the perfect angle to shoot a basketball into the ring, the article focused solely on determining the angle, ignoring other factors such as force, mass, and acceleration. The set-up, therefore, optimized to provide a medium of approximating the two sides of a right-angled triangle from the curve traced by the projectile that gets into the basket and calculating the side not possible to measure. Already developed trigonometric functions are then used to calculate the angle. However, from the assumptions made, this method has several limitations. Nevertheless, the approach provides a suitable foundation to introduce differential calculus in solving for the angles. This work is a field that can be explored in further research work. It is possible to find the optimum angle through which to shoot a basketball right into the basket. For players, achieving this angle may require a lot of practice to get the angles right inherently.

Cited Works

Brannan, David A et al. Geometry. Cambridge University Press, 2012.Burger, Edward B. Geometry. Holt Mcdougal, 2013.

"Trigonometry." The Penguin Dictionary Of Mathematics. Nelson, D, (ed.) 4th ed., Penguin, 2008.

Rusczyk, Richard. Introduction To Geometry. Aops Inc., 2016.

"Trigonometry." Khan Academy, https://www.khanacademy.org/math/algebra2/trig-functions/modal/a/trig-unit-circle-review. Accessed 14 Mar 2018.Zill, Dennis G, and Jacqueline M Dewar. Trigonometry. Jones & Bartlett Learning, 2012.