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The derivative of secx : fundamental geometric and algebraic properties of the secant trigonometric function

Nowadays, trigonometry is widely used in various fields of scientific study. However, it is a serious mistake to consider that those students, who have linked their lives with disciplines that are far from pure mathematics, should not understand the primal rules of trigonometry. In truth, this mathematical discipline finds its applications in different spheres of ordinary student’s life. Of course, one is not obliged to know how to solve a standard differential equation in order to compose an official personal statement or a literary essay. Nevertheless, various ordinary assignments require at least a superficial familiarity with the basic trigonometric principles and terms, such as the sine of the angle or the derivative of secx. In fact, the precise calculation of the trigonometric functions appertains to the class of highly specialized and time-consuming assignments. Fortunately, the rapid and ubiquitous development of computer technology permits most people to eschew this objective because of the widespread availability of specially designed mathematical software that supply users with accurate results of built-in trigonometric functions for any angle. Nevertheless, in order to obtain accurate results, one has to know how to formulate correct interpellation. Therefore, let us examine the fundamental properties of the basic trigonometric functions, using the derivative of secx as an example.

Suppose that x is a random number or angle. The value, which is the inverse of the cosine of the angle (number), is called secant of x and is denoted by sec x: sex x = 1/cos x. The function y = sec x is one of the fundamental trigonometric functions. Let us describe the basic mathematical properties of this function using the properties of the cosine function. From the definition of the secant implies that this function is periodic with the same period as the cosine (T = 2π). Therefore, for all integer values of n and all values of x in the domain secant the following equation is correct: sec (x + 2πn) = sec x. The cosine is an even function. Thereby, for all values of x in the domain of the secant the following equation is correct: sec (-x) = sec x. The secant is an unbounded function. The graph of the function y = sec x in the Cartesian coordinate system is called the secant curve.

The primal trigonometric functions: a concise introduction to trigonometry

The trigonometric functions are the basic mathematical functions. The argument of the trigonometric function is the angle. Nowadays, applications of the trigonometric functions are extremely diverse. Sociological statistics demonstrates that at least every tenth mathematical research proposal includes references to the scientific articles that are dedicated to various fields of modern trigonometry. For example, any batch processes can be expressed as the sum of trigonometric functions (Fourier series), thus, all types of researches in this sphere comprise trigonometric equations and specific calculations. In addition, these functions are widely used in various mathematical objectives that are dedicated to the solution of differential and functional equations. The geometric definition of the trigonometric functions can be easily formulated by using the unit circle (a circle of radius r – 1). In the standard unit circle the radius of the circle and the projection of the arbitrarily chosen point M on the circumference of a standard unit circle with the coordinates x, y form a right-angled triangle, wherein x and y are the catheti, and the radius of the circle is the hypotenuse. Using different trigonometric functions, we can describe various relationships between the parties and the acute angles in a standard right-angled triangle. We assume that the triangle lies in the Euclidean plane; therefore, the sum of the angles equals π. This statement means that the angle between the hypotenuse and the cathetus lies between 0 and π/2. Using the formula or definition through the standard unit circle, we can expand the scope of the definition of the trigonometric functions on the set of real numbers. Therefore, taking into account that the trigonometric functions are the mathematical functions of the angle, we can define the basic trigonometric functions as the ratio of the rectangular sides of the triangle or the length of certain segments in the unit circle. However, modern literary sources usually express trigonometric functions in terms of sums of the series or as solutions of certain differential equations. One can find a considerable amount of such examples in virtually all samples of a standard case study template. Thereby, this statement allows us to expand the domain of these functions on arbitrary real numbers and/or even complex numbers. Currently, there exist six fundamental trigonometric functions listed below together with equations relating them to each other. For the last four functions, the ratios are often called the definitions of these functions. However, we can define these features geometrically or using other mathematical methods. Here is a register of these functions, which comprises a brief description of each function:

  • The sine of the angle is the ratio of the opposite leg to the hypotenuse: sin a = a/c. This ratio is independent of the triangle ABC comprising an angle α because all these triangles are similar. The derivative of sin x: the derivative of f(x) = sin x is given by f '(x) = cos x.
  • The cosine of the angle can be determined as the ratio of the adjacent leg to the hypotenuse: cos a = b/c. In fact, sin b = b/c, therefore, the sine of an acute angle in the triangle is equal to the cosine of the second angle and vice versa. The derivative of cos x: the derivative of f(x) = cos x is given by f '(x) = -sin x.
  • The tangent of the angle is the ratio of the opposite leg to an adjacent: tg a = a/b. The derivative of tan x. The derivative of f(x) = tan x can be formulated using the following equation: f '(x) = sec2 x.
  • The cotangent of the angle is the ratio of the adjacent leg to an opposite leg: ctg a = b/a. The cotangent of the one acute angle in a right triangle is equal to the tangent of the second, and vice versa. The derivative of cot x: the derivative of f(x) = cot x is given by f '(x) = - csc2 x.
  • The secant of the angle is the ratio of the hypotenuse to the adjacent leg: sec a = c/b. The derivative of secx : the derivative of f(x) = sec x tan x is given by f '(x) = sec x tan x.
  • The cosecant of the angle is the ratio of the hypotenuse to the opposite leg: cosec a = c/a. The derivative of csc x: the derivative of f(x) = csc x is given by f '(x) = - csc x cot x.

The history of trigonometry: the main stages of development

In fact, the word ‘trigonometry’ is of Greek origin. Translated it means ‘measurement of triangles’. For the first time, the word ‘trigonometry’ was used in the title of the book of the German theologian and mathematician Pitiscus in 1505. Like all other branches of mathematics that originated in ancient times, trigonometry is the result of attempts to solve those problems with which man had to face in practice, such as different objectives connected with practical land surveying, architecture and astronomy. The first reliably attested trigonometric tables were compiled in the second century BC. Unfortunately, the original trigonometric tables were lost. Nevertheless, the famous Almagest (‘Great building’) written by Alexandrian astronomer Ptolemy comprises a considerable number of references and mentions about these works. Moreover, Almagest also contains many original conceptions that can be considered as results of the significant improvement and refinement of the original trigonometric conceptions. Thereby, Almagest includes formulas for the sum of sine and cosine of two angles. Furthermore, this book also comprises several elements of the spherical trigonometry. This mathematical treatise made a revolution in the field of geometry. Using modern terminology, we can say that this work was a serious grant proposal, which definitely created a furor among the scientific community of the time. During the Middle Ages, the greatest progress in the development of trigonometry has been achieved by differently distinguished mathematics of the Central Asia. Nevertheless, for a quite long period trigonometry was considered as an insignificant part of astronomy. One of the most notable works on trigonometry of that period is, doubtlessly, the Treatise on the quadrilateral written by Nasir al-Din al-Tusi (Khawaja Muhammad ibn Muhammad ibn Hasan Tusi). The next notable achievement in the history of trigonometry is connected with the name of the noted German scholar - Regiomontanus who published one of the most significant papers from the point of view of modern trigonometry (XV century). His main work: the Five books of different kinds of triangles contains a fairly complete account of the fundamental principles of trigonometry. In truth, this work differs from our current trigonometry textbooks only in the absence of convenient modern notation. After the appearance of the Five books of different kinds of triangles Regiomontanus trigonometry finally emerged as an acknowledged scientific discipline, independent of astronomy. Regiomontanus was also the first scientist who compiled detailed trigonometric tables. A significant contribution to the development of trigonometry was done by Leonhard Euler. In fact, he formulated virtually all modern definitions of trigonometric functions, such as the derivative of secx, cos x, sin x, etc., studied the main functions of an arbitrary angle and indicated the close connection of the trigonometric functions with exponential functions. Currently, the trigonometric functions appertain to the group of fundamental conceptions of a modern mathematical apparatus. They are closely related to the inverse functions. In addition, the trigonometric functions are also widely used in the so-called harmonic analysis, which is dedicated to study of different kinds of batch processes: oscillatory motion, wave propagation, etc.

The etymology of the trigonometric terms: concise references

 

Obviously, at the moment, we cannot track all first dates of use of different trigonometric and geometric terms with sufficient confidence. Unfortunately, even those mathematical works that have survived to our time were not written in the APA paper format. Thus, it is a very difficult assignment to create a reliable cross-reference system, which can indicate etymological roots of all these terms. However, we can determine the origin of the lion’s share of the fundamental mathematical concepts, eschewing possible misconceptions. For example, the etymology of the word ‘sine’ goes back to the Latin word ‘sinus’ that means ‘bay’. Firstly, this term (in its contemporary mathematical sense) was used in famous mathematical treatises, which were based on the Arabic mathematical works, in the XII century. The first use of the word ‘tangent’ can be traced to the Latin word ‘tangens’, which means ‘concerning’, because the line through which we define the tangent touches the circle of the unit radius. The term ‘secant’ also comes from the Latin language. The original Latin verb ‘secans’ means ‘slicing’ or ‘cutting’. This term indicates that the line metaphorically cuts the standard unit circle. In fact, Edmund Gunter firstly suggested those trigonometric functions whose names are formed using the prefix ‘co-’, such as the cotangent, cosecant and cosine, in the work Canon triangulorum, which was published in 1620. In fact, virtually all basic terms in the sphere of modern trigonometry, such as the tangent or the derivative of secx, as well as in the sphere of linear algebra, are of Latin origin.

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