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Journal articles on the topic 'TRIGONOMETRIC FUNCTION'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Akhmedova, Gavkharkhon, and Ozodakhon Makhmudova. "NON-STANDARD SOLUTIONS OF TRIGONOMETRIC EQUATIONS - TRIGONOMETRIK TENGLAMALARNI YECHISHNING NOSTANDART USULLARI." Nazariy va amaliy tadqiqotlar xalqaro jurnali 2, no. 2 (2022): 40–50. https://doi.org/10.5281/zenodo.6466296.

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This article presents a method for non-standard solutions of trigonometric equations. In particular, multiplication of both parts of the equation by the same trigonometric function, addition and subtraction of the same number or the same trigonometric function on both sides, proportions, elements of mathematical analysis, scalar product of vectors are used. -  Ushbu maqolada trigonometrik tenglamalarni yechishning nostandart usullari haqida ma’lumotlar berilgan. Jumladan tenglamaning har ikki tomonini bir hil trigonometrik funksiyaga ko‘paytirish, har ikki tomoniga bir xil son

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2

Martín Fernández, Enrique, Juan Francisco Ruiz-Hidalgo, and Luis Rico Romero. "Conversions Between Trigonometric Representation Systems by Pre-service Secondary School Teachers." PNA. Revista de Investigación en Didáctica de la Matemática 16, no. 3 (2022): 237–63. http://dx.doi.org/10.30827/pna.v16i3.21957.

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Understanding trigonometry relational system is a school mathemat-ics demanding topic. The angle, the unit circle and the trigonometric functions are its foundational notions. Trigonometric contents mean-ing and their understanding involve these three concepts and their re-lationships. This research aims to deepen in the pre-service teachers’ understanding about the angle, the unit circle and the trigonometric function when converting notions between two trigonometric repre-sentation systems based on the unit circle and the trigonometric func-tions. The results indicate that pre-service mathem

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3

Nitin, Darkunde, and Ghodechor Sanjay. "On Wilker's and Huygen's Type Inequalities for Generalized Trigonometric and Hyperbolic Functions." Indian Journal of Science and Technology 17, no. 9 (2024): 787–93. https://doi.org/10.17485/IJST/v17i9.3206.

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Abstract <strong>Objectives:</strong> The Trigonometric inequalities, generalized trigonometric inequalities which have been obtained by Wilker and Cusa Huygens have attracted attention of so many researchers. Generalized trigonometric functions are simple generalization of the classical trigonometric functions. It is related to the r- Laplacian, which is known as a non-linear differential operator. <strong>Method:</strong> For the establishment of inequalities involving generalized trigonometric and hyperbolic functions convexity plays the important role in many aspects, also M

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4

Putranti, Sri Rejeki Dwi. "Relationship between Trigonometry Functions with Hyperbolic Function." Aloha International Journal of Multidisciplinary Advancement (AIJMU) 1, no. 4 (2019): 82. http://dx.doi.org/10.33846/aijmu10402.

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Many engineering problems can be solved by methods involving complex numbers and complex functions. In the definitions below we will prove the relationship between trigonometric functions and hyperbolic functions, where the hyperbolic function is an extension of the trigonometric function. Keywords: trigonometric functions; hyperbolic functions

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5

Setiawan, Yayan Eryk. "PROSPECTIVE TEACHERS REPRESENTATIONS IN PROBLEM SOLVING OF SPECIAL ANGLE TRIGONOMETRY FUNCTIONS BASED ON THE LEVEL OF ABILITY." Infinity Journal 11, no. 1 (2022): 55. http://dx.doi.org/10.22460/infinity.v11i1.p55-76.

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One of the materials used as the basis for solving trigonometric function problems is special angle trigonometry. Prospective teachers' representation in problem-solving of trigonometric functions with special angles is thought to be influenced by prospective teachers' abilities. Therefore, this study aims to analyze the representations used by prospective teachers in problem-solving of special angle trigonometric function based on ability categories. This research is qualitative descriptive. The research subjects are prospective teachers of the mathematics education study program at a univers

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Saregar, Antomi. "Analisis Spektrum Energi dan Fungsi Gelombang Potensial Non-Centra Menggunakan Supersimetri Mekanika Kuantum." Jurnal Ilmiah Pendidikan Fisika Al-Biruni 4, no. 2 (2015): 193. http://dx.doi.org/10.24042/jpifalbiruni.v4i2.92.

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The objectives of this study were: 1) to describe the results of wave and energy function of the Non-central potential system of potential combinations of trigonometric Poschl-Teller plus Rosen Morse, Coloumb and OH 3D potential, and Rosen Morse trigonometric potential plus Pochl-Teller analyzed using Supersymmetry quantum mechanics (SUSY QM); 2) to know the visualization of wave function and energy level at point 1. This study is a literature study conducted from July 2013 to December 2015. Non-central potential of potential combinations of trigonometric Poschl-Teller potency plus Rosen Morse

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Saregar, Antomi. "Analisis Spektrum Energi dan Fungsi Gelombang Potensial Non-Central Menggunakan Supersimetri Mekanika Kuantum." Jurnal Ilmiah Pendidikan Fisika Al-Biruni 4, no. 2 (2015): 193. https://doi.org/10.24042/jipfalbiruni.v4i2.92.

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The objectives of this study were: 1) to describe the results of wave and energy function of the Non-central potential system of potential combinations of trigonometric Poschl-Teller plus Rosen Morse, Coloumb and OH 3D potential, and Rosen Morse trigonometric potential plus Pochl-Teller analyzed using Supersymmetry quantum mechanics (SUSY QM); 2) to know the visualization of wave function and energy level at point 1. This study is a literature study conducted from July 2013 to December 2015. Non-central potential of potential combinations of trigonometric Poschl-Teller potency plus Rosen Morse

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8

Baric, Mate, David Brčić, Mate Kosor, and Roko Jelic. "An Axiom of True Courses Calculation in Great Circle Navigation." Journal of Marine Science and Engineering 9, no. 6 (2021): 603. http://dx.doi.org/10.3390/jmse9060603.

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Based on traditional expressions and spherical trigonometry, at present, great circle navigation is undertaken using various navigational software packages. Recent research has mainly focused on vector algebra. These problems are calculated numerically and are thus suited to computer-aided great circle navigation. However, essential knowledge requires the navigator to be able to calculate navigation parameters without the use of aids. This requirement is met using spherical trigonometry functions and the Napier wheel. In addition, to facilitate calculation, certain axioms have been developed t

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9

Halim, Nabil Fachrudin, Moh Mahfud Effendi, and Mayang Dintarini. "Analysis of Trigonometry Learning Outcomes in the Application of Geogebra-Assisted Jigsaw Methods." Mathematics Education Journal 7, no. 1 (2023): 86–99. http://dx.doi.org/10.22219/mej.v7i1.23266.

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In the material of graphs of trigonometric functions, students are expected to be able to draw graphs of their functions which later affect student learning outcomes on the material of graphs of trigonometric functions. The purpose of this study was to analyze the learning outcomes of trigonometry in the application of Geogebra-assisted jigsaw learning. The data collection technique is done by giving test questions with trigonometric function graph material. The test results are used to determine the learning outcomes of the cognitive domain. To obtain learning outcomes in the affective and ps

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10

AGHELI, BAHRAM. "Approximate Solution of Bratu Differential Equations Using Trigonometric Basic Functions." Kragujevac Journal of Mathematics 45, no. 02 (2021): 203–14. http://dx.doi.org/10.46793/kgjmat2102.203a.

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In this paper, I have proposed a method for ﬁnding an approximate function for Bratu diﬀerential equations (BDEs), in which trigonometric basic functions are used. First, by deﬁning trigonometric basic functions, I deﬁne the values of the transformation function in relation to trigonometric basis functions (TBFs). Following that, the approximate function is deﬁned as a linear combination of trigonometric base functions and values of transform function which is named trigonometric transform method (TTM), and the convergence of the method is also presented. To get an approximate solution functio

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11

Titaley, Jullia, Tohap Manurung, and Henriette D. Titaley. "CUBIC AND QUADRATIC POLYNOMIAL ON JULIA SET WITH TRIGONOMETRIC FUNCTION." JURNAL ILMIAH SAINS 18, no. 2 (2018): 103. http://dx.doi.org/10.35799/jis.18.2.2018.21555.

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CUBIC AND QUADRATIC POLYNOMIAL ON JULIA SET WITH TRIGONOMETRIC FUNCTIONABSTRACTJulia set are defined by iterating a function of a complex number and is generated from the iterated function . We investigate in this paper the complex dynamics of different functions and applied iteration function system to generate an entire new class of julia set. The purpose of this research is to make variation of Cubic and Quadratic polynomial on Julia Set and the two obvious to investigate from julia set are Sine and Cosine function. The results thus obtained are innovative and studies about different behavi

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12

Darkunde, Nitin, and Sanjay Ghodechor. "On Wilker’s and Huygen’s Type Inequalities for Generalized Trigonometric and Hyperbolic Functions." Indian Journal Of Science And Technology 17, no. 9 (2024): 787–93. http://dx.doi.org/10.17485/ijst/v17i9.3206.

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Objectives: The Trigonometric inequalities, generalized trigonometric inequalities which have been obtained by Wilker and Cusa Huygens have attracted attention of so many researchers. Generalized trigonometric functions are simple generalization of the classical trigonometric functions. It is related to the r- Laplacian, which is known as a non-linear differential operator. Method: For the establishment of inequalities involving generalized trigonometric and hyperbolic functions convexity plays the important role in many aspects, also Monotonicity rule is used for sharpness of inequalities. Th

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13

Li, Bo, Yan Zhang, and Xiquan Liang. "Several Differentiation Formulas of Special Functions. Part III." Formalized Mathematics 14, no. 1 (2006): 37–45. http://dx.doi.org/10.2478/v10037-006-0006-z.

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Several Differentiation Formulas of Special Functions. Part III In this article, we give several differentiation formulas of special and composite functions including trigonometric function, inverse trigonometric function, polynomial function and logarithmic function.

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14

Medvegyev, Péter. "On the construction of the elementary trigonometric functions." Pure Mathematics and Applications 30, no. 3 (2022): 54–64. http://dx.doi.org/10.2478/puma-2022-0024.

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Abstract The article discusses the construction of the elementary trigonometric functions. It discusses several approaches, but the main message is that to construct the trigonometric functions one needs to follow the same approach as one should use during the construction of the real exponential function, but one should use complex numbers. The key point is that one must construct the complex square root function to define the trigonometric functions on the binary numbers and then one should use some convexity argument to prove their differentiability. This approach, as the power series appro

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Stewart, Seán M. "A look back at a long-forgotten trigonometric function: the versine function and its inverse." Mathematical Gazette 106, no. 565 (2022): 84–94. http://dx.doi.org/10.1017/mag.2022.13.

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Ask anyone who has studied mathematics to a moderate level how many trigonometric functions there are and one is likely to be presented with a range of answers depending on what the person being asked is most likely to remember. Perhaps the ‘calculator button’ three of sine, cosine, and tangent will come to mind as these are the three trigonometric functions found on any standard scientific calculator. At a stretch, perhaps the names for their respective reciprocals, cosecant, secant and cotangent, will be recalled. Beyond the modern standard six, looking at calculus or trigonometric texts pub

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CHAND, A. K. B., M. A. NAVASCUÉS, P. VISWANATHAN, and S. K. KATIYAR. "FRACTAL TRIGONOMETRIC POLYNOMIALS FOR RESTRICTED RANGE APPROXIMATION." Fractals 24, no. 02 (2016): 1650022. http://dx.doi.org/10.1142/s0218348x16500225.

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One-sided approximation tackles the problem of approximation of a prescribed function by simple traditional functions such as polynomials or trigonometric functions that lie completely above or below it. In this paper, we use the concept of fractal interpolation function (FIF), precisely of fractal trigonometric polynomials, to construct one-sided uniform approximants for some classes of continuous functions.

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HE, WEN-YU, and WEI-XIN REN. "ADAPTIVE TRIGONOMETRIC HERMITE WAVELET FINITE ELEMENT METHOD FOR STRUCTURAL ANALYSIS." International Journal of Structural Stability and Dynamics 13, no. 01 (2013): 1350007. http://dx.doi.org/10.1142/s0219455413500077.

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Owing to its good approximation characteristics of trigonometric functions and the multi-resolution local characteristics of wavelet, the trigonometric Hermite wavelet function is used as the element interpolation function. The corresponding trigonometric wavelet beam element is formulated based on the principle of minimum potential energy. As the order of wavelet can be enhanced easily and the multi-resolution can be achieved by the multi-scale of wavelet, the hierarchical and multi-resolution trigonometric wavelet beam element methods are proposed for the adaptive analysis. Numerical example

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Reynolds, Robert. "Extended Levett trigonometric series." PLOS One 20, no. 5 (2025): e0320045. https://doi.org/10.1371/journal.pone.0320045.

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An extension of two finite trigonometric series is studied to derive closed form formulae involving the Hurwitz-Lerch zeta function. The trigonometric series involves angles with a geometric series involving the powers of 3. These closed formulae are used to derive composite finite and infinite series involving special functions, trigonometric functions and fundamental constants. A short table summarizing some interesting results is produced.

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Fomin, Vasiliy I. "About complex operator functions of a complex operator variable." Russian Universities Reports. Mathematics, no. 147 (2024): 325–51. http://dx.doi.org/10.20310/2686-9667-2024-29-147-325-351.

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We consider a family of complex operator functions whose domain and range of values are included in the real Banach algebra of bounded linear complex operators acting in the Banach space of complex vectors over the field of real numbers. It is shown that the study of a function from this family can be reduced to the study of a pair of real operator functions of two real operator variables. The main elementary functions of this family are considered: power function; exponent; trigonometric functions of sine, cosine, tangent, cotangent, secant, cosecant; hyperbolic sine, cosine, tangent, cotange

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Zayed, E. M. E., and Khaled A. Gepreel. "A series of complexiton soliton solutions for nonlinear Jaulent—Miodek PDEs using the Riccati equations method." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 5 (2011): 1001–15. http://dx.doi.org/10.1017/s0308210510000405.

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We use the extended multiple Riccati equations expansion method to construct a series of double-soliton-like solutions, double triangular function solutions and complexiton soliton solutions for nonlinear Jaulent-Miodek PDEs. With the help of symbolic computation using Maple and Mathematica, we obtain many new types of complexiton soliton solutions, i.e. various combinations of trigonometric periodic functions and hyperbolic function solutions, various combinations of trigonometric periodic functions and rational function solutions, and various combinations of hyperbolic functions and rational

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21

Phan-Yamada, Tuyetdong, and Walter M. Yamada. "Exploroing Polar Curves with GeoGebra." Mathematics Teacher 106, no. 3 (2012): 228–33. http://dx.doi.org/10.5951/mathteacher.106.3.0228.

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Most trigonometry textbooks teach the graphing of polar equations as a two-step process: (1) plot the points corresponding to values of θ such as π, π/2, π/3, π/4, π/6, and so on; and then (2) connect these points with a curve that follows the behavior of the trigonometric function in the Cartesian plane. Many students have difficulty using this method to graph general polar curves. The difficulty seems to stem from an inability to convert changes in the value of the trigonometric equation as a function of angle (abscissa vs. ordinate in Cartesian coordinates) to changes of the radius as a fun

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Kazlouskaya, N. Yu, and Ya A. Rovba. "On the approximation of the | sin |s x function by rational trigonometric operators of the Fejér type." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 59, no. 2 (2023): 95–109. http://dx.doi.org/10.29235/1561-2430-2023-59-2-95-109.

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Approximation by trigonometric Fourier series is a well-developed branch of the theory of approximation by polynomials. Methods of approximation by rational trigonometric Fourier series have not been researched so deeply yet. In particular, rational trigonometric operators of the Fejér type have not been used in the rational approximation with free poles. In this paper, we consider the approximation of the function | sin | , (0;2), ∈ s x s by rational trigonometric operators of the Fejér type. An integral representation of the remainder for the above-mentioned approximation is obtained. An est

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Shah, Yogendra Prasad. "Taylor Series of Trigonometric Function for Lower Order Visualization." MATHunesa: Jurnal Ilmiah Matematika 12, no. 2 (2024): 352–57. https://doi.org/10.26740/mathunesa.v12n2.p352-357.

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Abstract About the simplest kind of functions are polynomials. Under addition, multiplication, integration, and differentiation, they are closed. A function can be roughly expressed in terms of polynomials of a certain degree using Taylor series. This provides a clear picture of the function's local behavior. The Taylor series fits the function at the place where it is computed more closely the more terms there are. In various fields of natural and social science, trigonometric functions are used. In this work, we examine the tailored series of trigonometric functions using Wolfram Mathematica

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Dattoli, Giuseppe, Emanuele Di Palma, and Silvia Licciardi. "On an Umbral Point of View of the Gaussian and Gaussian-like Functions." Symmetry 15, no. 12 (2023): 2157. http://dx.doi.org/10.3390/sym15122157.

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The theory of Gaussian functions is reformulated using an umbral point of view. The symbolic method we adopt here allows an interpretation of the Gaussian in terms of a Lorentzian image function. The formalism also suggests the introduction of a new point of view of trigonometry, opening a new interpretation of the associated special functions. The Erfi(x), is, for example, interpreted as the “sine” of the Gaussian trigonometry. The possibilities offered by the Umbral restyling proposed here are noticeable and offered by the formalism itself. We mention the link between higher-order Gaussian t

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Odhah, Omalsad Hamood, Huda M. Alshanbari, Zubair Ahmad, Faridoon Khan, and Abd Al-Aziz Hosni El-Bagoury. "A Novel Probabilistic Approach Based on Trigonometric Function: Model, Theory with Practical Applications." Symmetry 15, no. 8 (2023): 1528. http://dx.doi.org/10.3390/sym15081528.

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Proposing new families of probability models for data modeling in applied sectors is a prominent research topic. This paper also proposes a new method based on the trigonometric function to derive the updated form of the existing probability models. The proposed family is called the cotangent trigonometric-G family of distributions. Based on the cotangent trigonometric-G method, a new version of the Weibull model, namely, the cotangent trigonometric Weibull distribution, is studied. Certain mathematical properties of the cotangent trigonometric-G family are derived. The estimators of the cotan

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Bee, Kang Gyu, and Jeon Ye Won. "IMPLEMENTATION OF BIORHYTHM GRAPH WITH TRIGONOMETRIC FUNCTIONS USING PYTHON." MATTER: International Journal of Science and Technology 9, no. 1 (2023): 66–73. http://dx.doi.org/10.20319/mijst.2023.91.6673.

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This study aimed to implement the researcher's biorhythm graph using a programing language called Python in relation to trigonometric functions. The etymology of biorhythm is a combination of two Greek words, Bio, which means life, and Rhythm, which means regular and accurate rhythm, and means the rules of human life rhythm. In other words, the biorhythm is a theory that everyone is governed by three rhythm curves called physical rhythm, emotional rhythm, and intellectual rhythm that start inside the body from birth to death. During this study, biorhythm and trigonometric functions were studie

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Bee, Kang Gyu, and Jeon Ye Won. "Implementation of Biorhythm Graph with Trigonometric Functions using Python." MATTER: International Journal of Science and Technology 9 (July 15, 2023): 66–73. http://dx.doi.org/10.20319/mijst.2023.9.6673.

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This study aimed to implement the researcher's biorhythm graph using a programing language called Python in relation to trigonometric functions. The etymology of biorhythm is a combination of two Greek words, Bio, which means life, and Rhythm, which means regular and accurate rhythm, and means the rules of human life rhythm. In other words, the biorhythm is a theory that everyone is governed by three rhythm curves called physical rhythm, emotional rhythm, and intellectual rhythm that start inside the body from birth to death. During this study, biorhythm and trigonometric functions were studie

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Devine, M. L. "Real time trigonometric function evaluation." Microprocessors and Microsystems 16, no. 8 (1992): 417–25. http://dx.doi.org/10.1016/0141-9331(92)90028-r.

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Reynolds, Robert, and Allan Stauffer. "Extended Prudnikov sum." AIMS Mathematics 7, no. 10 (2022): 18576–86. http://dx.doi.org/10.3934/math.20221021.

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<abstract><p>A Prudnikov sum is extended to derive the finite sum of the Hurwitz-Lerch Zeta function in terms of the Hurwitz-Lerch Zeta function. This formula is then used to evaluate a number trigonometric sums and products in terms of other trigonometric functions. These sums and products are taken over positive integers which can be simplified in certain circumstances. The results obtained include generalizations of linear combinations of the Hurwitz-Lerch Zeta functions and involving powers of 2 evaluated in terms of sums of Hurwitz-Lerch Zeta functions. Some of these derivatio

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Li, Li, Jerito Pereira, and Neni Hermita. "Improving the Trigonometric Functions Learning Concept with Dynamic Mathematics Software." International Journal of Scientific Research and Management 10, no. 04 (2022): 386–96. http://dx.doi.org/10.18535/ijsrm/v10i4.m01.

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Since the 21st century, the new model of Internet+Education has gradually penetrated into the field of mathematics education and has become a hot topic in the current research on mathematics education, and education informatization has become the main development trend. Therefore, the purpose of this research is to help students and teachers how to use Hawgent dynamic mathematics software to help students learn trigonometric functions and help teachers practice trigonometric functions. The research method used is the research and development (R&D) method and this research was done at Chong

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Yessenbayeva, Gulsim A., Gulmira A. Yessenbayeva, A. T. Kasimov, and N. K. Syzdykova. "On the boundedness of the partial sums operator for the Fourier series in the function classes families associated with harmonic intervals." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 103, no. 3 (2021): 131–39. http://dx.doi.org/10.31489/2021m3/131-139.

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The article is devoted to the study of some data from the theory of functions approximation by trigonometric polynomials with a spectrum from special sets called harmonic intervals. Due to the limited perception range of devices, the perception range of the senses of the person himself, when studying a mathematical model it is often enough to find an approximation of the object so that the error (noise, interference, distortion) is outside the interval of perception. Harmonic intervals model problems of this kind to some extent. In the article the main components of the approximation theory of

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Zhengrong, Liu, Jiang Tianpei, Qin Peng, and Xu Qinfeng. "Trigonometric Function Periodic Wave Solutions and Their Limit Forms for the KdV-Like and the PC-Like Equations." Mathematical Problems in Engineering 2011 (2011): 1–23. http://dx.doi.org/10.1155/2011/810217.

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We use the bifurcation method of dynamical systems to study the periodic wave solutions and their limit forms for the KdV-like equationut+a(1+bu)uux+uxxx=0, and PC-like equationvtt- vttxx- (a1v+a2v2+a3v3)xx=0, respectively. Via some special phase orbits, we obtain some new explicit periodic wave solutions which are called trigonometric function periodic wave solutions because they are expressed in terms of trigonometric functions. We also show that the trigonometric function periodic wave solutions can be obtained from the limits of elliptic function periodic wave solutions. It is very interes

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Fernando Tercero Vitola de la Rosa. "Enseñanza y aprendizaje de la trigonometría: Un abordaje desde las investigaciones doctorales en educación matemática." GACETA DE PEDAGOGÍA, no. 45 (April 30, 2023): 228–53. http://dx.doi.org/10.56219/rgp.vi45.1900.

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La trigonometría es un área fundamental de la matemática con muchas dificultades en los procesos de aprendizaje y enseñanza. Este artículo describe el estado actual de las investigaciones doctorales recientes sobre didáctica de la trigonometría, analizando los vacíos existentes e identificando posibles temas de investigación. Se realizó una revisión sistemática de 18 tesis doctorales relacionadas con los descriptores trigonometría y funciones trigonométricas, obtenidas mediante buscadores académicos y repositorios de diferentes universidades existentes en internet, utilizando como categorías d

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Chii-Huei, Yu. "A Fractional Integral Involving Fractional Trigonometric Function." International Journal of Recent Research in Interdisciplinary Sciences (IJRRIS) 10, no. 3 (2023): 21–24. https://doi.org/10.5281/zenodo.8176924.

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<strong>Abstract:</strong> In this paper, based on Jumarie type of Riemann-Liouville (R-L) fractional calculus and a new multiplication of fractional analytic functions, we study a fractional integral involving fractional trigonometric function. In fact, our result is a generalization of traditional calculus result. <strong>Keywords:</strong> Jumarie type of R-L fractional calculus, new multiplication, fractional analytic functions, fractional integral, fractional trigonometric function. <strong>Title:</strong> A Fractional Integral Involving Fractional Trigonometric Function <strong>Author:</

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Chu, Wenchang. "Trigonometric Sums and Riemann Zeta Function." Mathematica Slovaca 73, no. 3 (2023): 595–612. http://dx.doi.org/10.1515/ms-2023-0044.

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ABSTRACT Several trigonometric sums are examined by means of partial fraction decompositions. Their generating functions will be evaluated in closed forms and the related asymptotic formulae in terms of Riemann zeta function will be established.

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Anastassiou, George A., and Seda Karateke. "Parametrized Half-Hyperbolic Tangent Function-Activated Complex-Valued Neural Network Approximation." Symmetry 16, no. 12 (2024): 1568. http://dx.doi.org/10.3390/sym16121568.

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In this paper, we create a family of neural network (NN) operators employing a parametrized and deformed half-hyperbolic tangent function as an activation function and a density function produced by the same activation function. Moreover, we consider the univariate quantitative approximations by complex-valued neural network (NN) operators of complex-valued functions on a compact domain. Pointwise and uniform convergence results on Banach spaces are acquired through trigonometric, hyperbolic, and hybrid-type hyperbolic–trigonometric approaches.

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Chii-Huei, Yu. "A Study on the Fractional Differential Problem of Some Type of Fractional Trigonometric Function." International Journal of Interdisciplinary Research and Innovations 12, no. 4 (2024): 21–25. https://doi.org/10.5281/zenodo.13949406.

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<strong>Abstract:</strong> In this paper, we obtain the formula of arbitrary order fractional derivative of some type of fractional trigonometric function by using the fractional Fourier series theory. Jumarie type of Riemann-Liouville (R-L) fractional derivative and a new multiplication of fractional analytic functions play important roles in this article. In fact, our result is a generalization of the result in classical calculus.  <strong>Keywords:</strong> Fractional derivative, fractional trigonometric function, fractional Fourier series, Jumarie type of R-L fractional derivative, ne

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Awaad, Bushra Kadhum, and Eman Samer Bhaya. "Trigonometric approximation of signals functions in Lipchitz spaces of order q ∈ N." Journal of Interdisciplinary Mathematics 28, no. 1 (2025): 109–15. https://doi.org/10.47974/jim-1801.

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Many researchers study the approximation of continuous functions using algebraic polynomials. But sometimes we demanded that the approximation is trigonometric function. Here we focus on a saturation problem on Lipchitz spaces of order q ∈ N and study the trigonometric approximation on that space.

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Kong, Sixiao, Chunbiao Li, Haibo Jiang, Yibo Zhao, and Yanling Wang. "Asymmetry Evolvement and Controllability of a Symmetric Hyperchaotic Map." Symmetry 13, no. 6 (2021): 1039. http://dx.doi.org/10.3390/sym13061039.

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Trigonometric functions were used to construct a 2-D symmetrical hyperchaotic map with infinitely many attractors. The regime of multistability depends on the periodicity of the trigonometric function, which is closely related to the initial condition. For this trigonometric nonlinearity and the introduction of an offset controller, the initial condition triggers a specific multistability evolvement, in which infinitely countless symmetric and asymmetric attractors are produced. Initial condition-triggered offset boosting is explored, combined with constant controlled offset regulation. Furthe

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Chen, Hongwei. "On some trigonometric power sums." International Journal of Mathematics and Mathematical Sciences 30, no. 3 (2002): 185–91. http://dx.doi.org/10.1155/s0161171202007731.

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Using the generating function method, the closed formulas for various power sums of trigonometric functions are established. The computer algebra system Maple is used to carry out the complex calculations.

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Yang, Yun Jie, and Yun Mei Zhao. "Derivation of Exact Solutions for the (3+1)-Dimensional KP Equation with (G'/G²)-Expansion Method." Advanced Materials Research 1044-1045 (October 2014): 1110–12. http://dx.doi.org/10.4028/www.scientific.net/amr.1044-1045.1110.

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Using the-expansion method proposed recently the exact solutions of the (3+1)-dimensional KP equation are found out. The exact solutions are expressed by two types of functions which are the trigonometric functions and the rational functions. When the parameters are taken as special values the trigonometric function solutions can be expressed as periodic wave solutions.

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Li, Bo, and Peng Wang. "Several Differentiation Formulas of Special Functions. Part IV." Formalized Mathematics 14, no. 3 (2006): 109–14. http://dx.doi.org/10.2478/v10037-006-0013-0.

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Several Differentiation Formulas of Special Functions. Part IV In this article, we give several differentiation formulas of special and composite functions including trigonometric function, polynomial function and logarithmic function.

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Omer, Faruk CETIN. "Students perceptions and development of conceptual understanding regarding trigonometry and trigonometric function." Educational Research and Reviews 10, no. 3 (2015): 338–50. http://dx.doi.org/10.5897/err2014.2017.

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Marufi, Marufi, Muhammad Ilyas, Muhammad Ikram, Rosidah Rosidah, and Phimlikid Kaewhanam. "Exploration of high school students' reasoning in solving trigonometric function problems." Al-Jabar : Jurnal Pendidikan Matematika 13, no. 2 (2022): 231–49. http://dx.doi.org/10.24042/ajpm.v13i2.12972.

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Reasoning has been extensively studied by many experts. However, Research on student reasoning in trigonometric problem solving, particularly those related to logical thinking skills is still sorely needed. This study aimed to explore students' reasoning in solving trigonometric function problems regarding logical thinking skills. The research was conducted using a qualitative approach. The research subjects involved high school students in Palopo, Indonesia. Based on the logical ability test results, three subjects were selected, namely students with high, medium, and low logical abilities. R

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Endaryono, Endaryono, and Wirda Hidayah Azzahra. "PENERAPAN KURVA TRIGONOMETRI UNTUK MOTIF KAIN BATIK - MEDIA PEMBELAJARAN PERSAMAAN TRIGONOMETRI." Jurnal Edukasi dan Sains Matematika (JES-MAT) 8, no. 1 (2022): 1–16. http://dx.doi.org/10.25134/jes-mat.v8i1.5249.

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The subject of trigonometric equations is a learning material given to high school students majoring in Mathematics and Natural Sciences and also deepened in student lectures in the fields of science such as mathematics, engineering, informatics and others. The problem discussed in this study is that learning media is still needed to support learning trigonometric equations. The purpose of this research is to create a media for visual simulation of the results of trigonometric equation functions, namely the sine function with various equation models and by modifying the coefficient values in t

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Hsu, Kuan-Lun. "Synthesis of trigonometric motion programs with lower motion characteristics." Mechanical Sciences 13, no. 1 (2022): 111–21. http://dx.doi.org/10.5194/ms-13-111-2022.

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Abstract. Motion characteristics are dimensionless peak values of velocity, acceleration, jerk, and acceleration multiplied by velocity of a motion program. In general, these peak values of synthesized motion programs should be as low as possible. Some trigonometric motion programs are widely used because they have a good compromise of all motion characteristics. A property in common for trigonometric motion programs is that their acceleration functions can be expressed as a qualitative shape of a sinusoidal function. The interval of the sinusoidal function is divided into several zones having

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Stavek, Jiri. "On the Hidden Beauty of Trigonometric Functions." Applied Physics Research 9, no. 2 (2017): 57. http://dx.doi.org/10.5539/apr.v9n2p57.

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In the unit circle with radius R = E0 = mc2 = 1 we have defined the trigonometric function cos(Theta) = v/c. The known trigonometric functions revealed the hidden relationships between sensible energy, latent energy, sensible momentum and latent momentum of the moving object, and the absorbed momentum from outside and the available momentum in the outside of the moving object. We present the trigonometric concept inspired by the old Babylonian clay tablet IM 55357 and based on the knowledge of the School of Athens (the fresco of Raphael) and the work of many generations of the Masters of trigo

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Li, Mei, and Wanqiang Shen. "Integral method from even to odd order for trigonometric B-spline basis." AIMS Mathematics 9, no. 12 (2024): 36470–92. https://doi.org/10.3934/math.20241729.

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<p>The conventional trigonometric B-spline basis of odd order for piecewise trigonometric polynomial space possesses a lot of good modeling properties. However, its order cannot be increased by the integral method like B-spline because of the particularity of the trigonometric polynomials. In the paper, a basis in an even-order trigonometric polynomial space is defined, and its integral relation with the existing odd-order trigonometric B-spline basis is obtained. First, the condition of the knot sequence is improved to ensure the nonnegativity of the prior odd-order trigonometric B-spli

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S, Chinthamani, and Lokesh P. "The Extension of Chebyshev Polynomial Bounds Involving Bazilevic Function." Indian Journal of Science and Technology 16, no. 27 (2023): 2040–46. https://doi.org/10.17485/IJST/v16i27.icrms-207.

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Abstract <strong>Objectives:</strong> To propose a new class of bi-univalent function based on Bazilevic Sakaguchi function using the trigonometric polynomials Tn ( q;eiq ) and to find the Taylor – Maclaurin coefficient inequalities and Fekete – Szego inequality for upper bounds. <strong>Methods:</strong> The Chebychev’s polynomial has vast applications in GFT. The powerful tool called convolution (Or Hadamard product), subordination techniques are used in designing the new class. In establishing the core results, derivative tests, triangle inequality and appro

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Goodman, T. N. T., and A. Sharma. "Trigonometric interpolation." Proceedings of the Edinburgh Mathematical Society 35, no. 3 (1992): 457–72. http://dx.doi.org/10.1017/s0013091500005745.

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We consider interpolation at 2n equidistant nodes in [0,π) from the space ℱN spanned by sines and cosines of odd multiples of x. This interpolation problem is shown to be correct for an arbitrary sequence of derivatives specified at all the nodes. Explicit expressions for the fundamental polynomials are obtained and it is shown that under mild smoothness assumptions on the function f interpolant from ℱN converges uniformly to f as the node spacing goes to zero.

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