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Inverse trig functions : an introduction to the field of trigonometry

We can say that two functions f and g are mutually inverse in case the equation y = f(x) is true simultaneously with the equation x = g(y). What are the main properties of inverse trig functions ?

  • 1. It is always either f(x(y)) = y or y(f (x)) = x.
  • 2. Also always remember that D(f) = E(g) and E(f) = D(g).
  • 3. If function f increases, then function g increases too, and vice versa – decreasing of function f goes alongside with decreasing of function g.
  • 4. The graphs are always symmetric regarding the line y = x.
  • 5. The derivative has the following property: g'(x) = 1/f'(g(x)).

First and foremost, what is mathematics? Many mathematicians, logicians and philosophers give definitions which cannot be found satisfactory, especially for those who are laymen in mathematics. We will not try to make the definition better. It is difficult to determine and single out completely even one branch of mathematics. The term ‘geometry’ is familiar to everyone, but not everyone knows its value. This word, especially in the past, was often used as a synonym for mathematics, and geometry and trigonometry were mentioned in the same sense in which we talk about mathematics now. Students can employ famous words of professor Veblen who was an outstanding geometer of Princeton University for their case study topics : he illustriously defined geometry as the part of mathematics that should be named with the very word ‘geometry’ according to the majority of expert voices. Nowadays, many mathematicians consider geometry and trigonometry as branches of algebra. But the very notion of ‘algebra’ has undergone a pronounced change during the last century. For many years, there was a discipline known as ‘algebra’, about the content of which there were no doubts. Then it swallowed a large part of the theory of invariants and the theory of groups, thus turning into ‘higher algebra’ or simply ‘algebra’. The considerable development of this discipline over the last thirty-five years has led to the emergence of the term ‘abstract algebra’ or ‘modern algebra’. Today, even a person with the most advanced knowledge of inverse trig functions cannot predict for how long we will have this ‘modern/abstract’ division.

Inverse trig functions : the most valuable pieces of the understanding

Since the seventeenth century the trigonometric functions began to be applied to the solution of equations, problems of mechanics, optics, electricity, radio engineering, also to describe oscillatory processes, propagation of waves, movement of various mechanisms, for the study of alternating electric current, and so on. Therefore, trigonometric functions were and still are comprehensively and deeply studied, and represent great importance for the whole of mathematics. The analytical theory of trigonometric functions was largely created in the eighteenth century by the outstanding mathematician and the member of the St. Petersburg Academy of Sciences whose name was Leonhard Euler – certainly, you already have met this name more than once provided you are a frequent visitor of this best resume writing service. The great scientific heritage of Euler includes excellent results relating to mathematical analysis, geometry, number theory, mechanics and other branches of the mathematical science. After Euler’s definitions of trigonometric functions the trigonometry science entered an unheard-of phase where various facts were proved by formal application of trigonometry formulas and evidence were much simpler, thus allowing a scientist to examine a function from a new angle. Thus, just then trigonometry evolved from the science of solving triangles into the science of trigonometric functions. In later development, the part of trigonometry which studies properties of trigonometric functions and relationships between them acquired the name goniometry (in translation: the science of measurement of angles, from the Greek gwnia – angle and metrew – measure).

Mathematics has so many different aspects that giving it a definition is as difficult as to formulate a single criteria for the classification of living organisms as animals or plants. Sometimes it is impossible to say with certainty whether the data of theorems, inverse trig functions and conclusions from them belong to mathematics or whether they belong to logic. Probably, it will be clearer if we say that mathematics is likely to begin where there is the concept of number – either in an explicit or implicit form and in some cases the concept could be so hidden that it is possible to detect it only after due consideration. The concept of number may has a different form, for example, when counting is being applied, or as a means of quantitative assessment of material objects, or as an abstraction of various dimensions, or, finally, as a concept related to some phenomena of nature. For instance, geometry has its source in problems, with which people encountered in everyday land measurement. It was natural human curiosity that aroused the desire to have a reliable logical foundation for all the empirical results which were well known from routine activities. Further reflections on this topic led to the emergence of Greek geometry.

Scientists are observing lots and lots of considerable difficulties in the study of the most developed areas of mathematics. Many of the most important and vital works accomplished over the past fifty years are based on new ideas, and new ideas, as opposed to inverse trig functions, are always very difficult to understand. When you meet them for the first time – for instance, while writing a synthesis essay – such ideas seem inexplicably distant, elusive and strange. They can differ so greatly from our usual way of thinking that you feel totally confused. All that you can do in this case is to read and re-read, and, gradually, dim outlines that seem entirely devoid of reality and even content are beginning to take a more definite shape, and all the strangeness disappears due to repeated exposure; finally, we are able to observe the thing that hides in a new idea, its value and its strength and even its great simplicity embedded in a variety of arguments.

Inverse trig functions : a brief history of the making of trigonometry science

Trigonometry is a Greek word and literally means the measurement of triangles (you should always use interpretations of Greek terms for your thesis format ). In this case, the measurement should be understood as finding a solution for triangles, i.e. defining the sides, angles, and other elements of the triangle provided the data about some of them are available. A large number of practical problems as well as problems of plane geometry, solid geometry, astronomy, and other branches of mathematical science lead to the problem of solving triangles. The emergence of trigonometry is associated with land surveying, astronomy and construction business. Although the name of the science arose relatively recently, many concepts and facts that are now attributable to trigonometry have been known, presumably, for two thousand years. For the first time, methods of solving triangles based on the relationships between the sides and angles of a triangle were found by the ancient Greek astronomer Hipparchus (the second century BC) and Claudius Ptolemy (the second century AD). Later, rules applied to the relationships between the sides of a triangle and its angles were called the trigonometric functions (do not confuse with inverse trig functions). Predominantly, the most significant contribution to the development of trigonometry was made by the Arab scientists Al-Batani (850-929) and Abu al-Wafa as well as by Mohamed Bin Mohammed (940-998) who compiled tables of sines and tangents with a granularity of 10' and to within 1/604. The law of sines was already known to the Indian scientist Bhaskara (born 1114, year of death unknown) and the Azerbaijani astronomer and mathematician Mohammed Nasireddin Tusi (1201-1274). In addition, Nasireddin Tusi successfully commits to paper both plane and spherical trigonometry as independent disciplines in his "Treatise on the complete quadrilateral".

The concept of sine also has a long and interesting history. In fact, different relationships of a triangle’s segments and a circle were found in the third century BC in the works of the great mathematicians of ancient Greece: Euclid, Archimedes, Apollonius of Perga. Essentially, we can find trigonometric functions and origins of inverse trig functions in these works too. In the Roman period, these relations were systematically studied by Menelaus (the first century BC), although they did not acquire a special name. The modern concept of sine a, for example, was studied as the chord of a doubled arc or as a semichord, which supports the central angle with the quantity a. In the fourth and fifth centuries, a special term appeared in the astronomy works by the great Indian scholar Aryabhata whose name was also given to the first Indian satellite. He gave the name ardhadzhiva (Ardha – half; Jiva - bowstring, which resembles a chord) to the standard segment that connects two ends of a chord. Usually, you can use various dissertation editing services in case you do not know how to interpret the terminology with Latin, Greek or Arabian roots. In the ninth century, Arabian mathematicians replaced the name by the Arabic word jibe (bulge). It was replaced by the Latin name sine (sinus – bending, curvature) during the translation of Arabic mathematical texts in later centuries.

The word ‘cosine’ is much younger. Cosine is an abbreviation of the Latin expression that means ‘the sine of the complementary or co-angle’, i.e. "extra sine" (cos a = sin (90 ° - a). Tangents have arisen in connection with inverse trig functions and the solution of the problem of finding the length of a shaded area. Tangent (as well as the cot) was coined in the tenth century by the Arab mathematician Abul-Vafa who first invented the table for finding tangents and cotangents. However, these discoveries have long remained unknown to European scientists; it was only the fourteenth century where tangents were rediscovered by the German mathematician and astronomer Regimontanom (died 1467). He was the first scientist who proved the theorem of tangents. Regiomontanus also made expanded trigonometric tables. Thanks to his works plane and spherical trigonometry became an independent and well-known discipline in Europe. Specifically speaking, the name "tangent" itself appeared in 1583 as a derivation from the Latin word tanger (touch). We can reveal the further development of trigonometry in the works by the outstanding astronomer Nicolaus Copernicus (1473-1543) who was a creator of the heliocentric system of the world. Other scientists who had trigonometry on its feet were Tycho Brahe (1546-1601), Johannes Kepler (1571-1630), and the mathematician Francois Vieta (1540-1603) who managed to solve completely the problem of calculating all the elements of a triangle (as well as inverse trig functions) from three known quantities. For a long time trigonometry has a purely geometric character, in other words, facts and calculations, which we now formulate in terms of trigonometric functions, were formulated and proved, basically, with the help of geometrical concepts and statements. That situation was somewhat reminiscent of the Middle Ages, however, analytical methods were used sometimes, especially after the origination of logarithms. Perhaps, the greatest stimulus to the development of trigonometry arose in connection with the solution of problems of astronomy, which were of great practical interest in those days (for example, to solve the problem of determining the location of a ship, predicting eclipses, etc.) Astronomers were deeply interested in the relationships between the sides and angles of spherical triangles. And it should be noted that mathematics of antiquity coped successfully with such tasks.

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