International Research Journal of Engineering and Technology (IRJET)
e-ISSN: 2395-0056
Volume: 11 Issue: 08 | Aug 2024
p-ISSN: 2395-0072
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Introducing Hidden Order Derivatives: Sine and Cosine Enjoying their Oscillation Rudranil Sahu1 1Department Of Mathematics, Burdwan Raj College, The University of Burdwan, West-Bengal, India
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Abstract- Sine and cosines are well known to be
problems. The well-known Fourier series depends upon trigonometric representations such as sines and cosines. The basic objective of such a representation is to disintegrate periodic functions into well-known sine and cosine components [2]. This is very easy because of the use of elementary trigonometric ratios. In addition, in this study, we adopted a new convention for further exploration. We have explored, ‘relation within the derivatives’ of sine and cosine functions in some particular order. In later sections, we present some results. We expect our methodology to penetrate curious minds, for a bright avenue in mathematics.
oscillatory functions. They are oscillating between (-1) and 1 for all such values permissible into sines and cosines. These elementary plane trigonometric ratios are appropriate for each other. The oscillation of sine and cosine functions is a pivotal concept with broadened applications in various fields. This versatility makes them essential tools for observing and analyzing periodic phenomena. Here, we introduce our proposal for new conventions related to the hidden order of the derived function. In this article, we have developed very elementary level fundamentals to expand this field in the future. This study explores the relationship between higher order derivatives of sines and cosines. At the beginning, it seems fairly obvious, but it has underlying mysteries. Our methodology addresses new notations, per the convenience of derivations. Through our outcomes, we seek to unfold other properties of periodic functions, ‘hidden’ beyond sines and cosines. Later, we conclude our work with an attempt to explore the properties of functions other than sines and cosines. Our work on such periodic entities represents a new avenue to the field of modern mathematics.
2. Intuitive preliminaries For the general readers, we have listed some of the basic preliminaries in this section. Here, our objective is to demonstrate a very crystal-clear introduction of the periodicity of function.
2.1 Functions Let be two nonvoid sets then a from set to set is a subset of . Thus, a relation R from ) A to B if the ordered pair ( in that case, it is abbreviated as ‘ is related to by ’. In accordance with these requirements, there are distinct types of relations, such as universal, void, identity, reflexive, symmetric, anti-symmetric and transitive. The foundations and allied consequences of function are tremendously important in mathematics and other disciplines as well. For two nonvoid sets (i.e., a subset of is called a function or mapping or a map) from if the following is maintained:
Key Words: Oscillatory function, Derived function, Plane trigonometric ratios. 1. Introduction Trigonometry focuses on the interesting story of triangles and their corresponding arms and angles. The history of ‘numerical relations of triangles’ found in ‘Ancient Egypt’ and ‘Babylonia’ paved the way for ‘early trigonometry’. However, the trigonometry we use today is found in ancient Greece. In addition to Egypt, Babylonia and Greece made contributions to trigonometry, Indian mathematicians such as AryabhataI also performed many works on this topic [3]. Initially, this inspired many applications, such as real-life approaches to problems corresponding ‘Height & Distance’, and problems related to triangles. Sine, Cosine, Tangent, Cosecant, Secant and Cotangent are six ratios that fall under the plane trigonometric function [5]. Currently, the foundation of trigonometry is very strong, and mathematicians are making great innovations in this field. Trigonometric functions are easy to use as sine and cosine functions are bounded functions in their domain, and their use can be helpful in solving complex
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1. 2.
For each a ,b ) If ( and(
such that( ) , then
)
Hence, we write , where is called the domain and where is the codomain of the function When the values of B are associated with real numbers, the corresponding function is said to be a real-valued function. Thus, we can infer that the functions are special kinds of ‘relations’ with the abovementioned inherent property. Depending upon the requirements, functions are classified into one-one, many-one, onto and into functions. The trigonometric functions given in this
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