scholarly journals Applying Computer Algebra Systems in Approximating the Trigonometric Functions

2018 ◽  
Vol 23 (3) ◽  
pp. 37 ◽  
Author(s):  
Le Quan ◽  
Thái Nhan
Keyword(s):  
Error Analysis ◽  
Computer Algebra ◽  
Numerical Algorithms ◽  
High Accuracy ◽  
Computer Algebra Systems ◽  
Trigonometric Functions ◽  
Modern Computer ◽  
Sine And Cosine Functions ◽  
Cosine Functions ◽  
Very High

We propose numerical algorithms which can be integrated with modern computer algebra systems in a way that is easily implemented to approximate the sine and cosine functions with an arbitrary accuracy. Our approach is based on Taylor’s expansion about a point having a form of kp, k∈Z and p=π/2, and being chosen such that it is closest to the argument. A full error analysis, which takes advantage of current computer algebra systems in approximating π with a very high accuracy, of our proposed methods is provided. A numerical integration application is performed to demonstrate the use of algorithms. Numerical and graphical results are implemented by MAPLE.

scholarly journals ESSENTIAL TRIGONOMETRY WITHOUT GEOMETRY

2019 ◽  
Vol 71 (1) ◽  
Author(s):  
John Gresham ◽  
Bryant Wyatt ◽  
Jesse Crawford
Keyword(s):  
Unit Circle ◽  
Elementary Level ◽  
Comprehensive Treatment ◽  
Advanced Mathematics ◽  
Trigonometric Functions ◽  
Geometric Constructions ◽  
Rigorous Approach ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Abstract The development of the trigonometric functions in introductory texts usually follows geometric constructions using right triangles or the unit circle. While these methods are satisfactory at the elementary level, advanced mathematics demands a more rigorous approach. Our purpose here is to revisit elementary trigonometry from an entirely analytic perspective. We will give a comprehensive treatment of the sine and cosine functions and will show how to derive the familiar theorems of trigonometry without reference to geometric definitions or constructions. Supplemental material is available for this article online.

scholarly journals Relation of Some Known Functions in terms of Generalized Meijer G -Functions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Syed Ali Haider Shah ◽  
Shahid Mubeen ◽  
Gauhar Rahman ◽  
Jihad Younis
Keyword(s):  
Bessel Functions ◽  
Logarithmic Function ◽  
Trigonometric Functions ◽  
Modified Bessel Functions ◽  
Modified Bessel Function ◽  
Exponential Functions ◽  
Hyperbolic Functions ◽  
Binomial Expansion ◽  
Sine And Cosine Functions ◽  
Cosine Functions

The aim of this paper is to prove some identities in the form of generalized Meijer G -function. We prove the relation of some known functions such as exponential functions, sine and cosine functions, product of exponential and trigonometric functions, product of exponential and hyperbolic functions, binomial expansion, logarithmic function, and sine integral, with the generalized Meijer G -function. We also prove the product of modified Bessel function of first and second kind in the form of generalized Meijer G -function and solve an integral involving the product of modified Bessel functions.

Application of modern computer algebra systems in food formulations and development: A case study

2017 ◽  
Vol 64 ◽  
pp. 48-59 ◽  
Author(s):  
Olga Musina ◽  
Predrag Putnik ◽  
Mohamed Koubaa ◽  
Francisco J. Barba ◽  
Ralf Greiner ◽  
...  
Keyword(s):  
Computer Algebra ◽  
Computer Algebra Systems ◽  
Modern Computer

scholarly journals Numerical Algorithms for Computing Eigenvalues of Discontinuous Dirac System Using Sinc-Gaussian Method

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. M. Tharwat ◽  
A. Al-Fhaid
Keyword(s):  
Error Analysis ◽  
Numerical Algorithms ◽  
High Accuracy ◽  
Transmission Conditions ◽  
Numerical Results ◽  
New Method ◽  
Dirac System ◽  
Sinc Method ◽  
Dirac Systems ◽  
Gaussian Method

The eigenvalues of a discontinuous regular Dirac systems with transmission conditions at the point of discontinuity are computed using the sinc-Gaussian method. The error analysis of this method for solving discontinuous regular Dirac system is discussed. It shows that the error decays exponentially in terms of the number of involved samples. Therefore, the accuracy of the new method is higher than the classical sinc-method. Numerical results indicating the high accuracy and effectiveness of these algorithms are presented. Comparisons with the classical sinc-method are given.

A New Approach to Obtaining Closed-Form Solutions for Higher Order Tetrahedral Finite Elements Using Modern Computer Algebra Systems

2011 ◽  
Author(s):  
Sara E. McCaslin ◽  
Panos S. Shiakolas ◽  
Brian H. Dennis ◽  
Kent L. Lawrence
Keyword(s):  
Parallel Processing ◽  
Closed Form ◽  
Computer Algebra ◽  
High Order ◽  
Computer Algebra Systems ◽  
Element Analysis ◽  
New Approach ◽  
Closed Form Solutions ◽  
Modern Computer ◽  
Stiffness Matrices

Closed-form solutions for straight-sided tetrahedral element stiffness matrices used in finite element analysis have been proven more efficient than numerically integrated solutions. These closed-form solutions are symbolically integrated using computer algebra systems such as Mathematica or Maple. However, even with memory and processing speed available on desktop computers today, major hindrances exist when attempting to symbolically evaluate the stiffness matrices for high order elements. This research proposes a new approach to obtaining closed-form solutions. Results are presented that demonstrate the feasibility of obtaining the stiffness matrices for high order tetrahedral elements through p-level 9 by use of parallel processing tools in Mathematica 7. Comparisons are made between serial and parallel approaches based on memory required to generate a solution. The serial approach requires more memory and can only generate closed-form solutions up to 7th order. The parallel processing approach presented requires less memory and can generate solutions up to 9th order.

Empirical Similarity Analysis Using Adaptive Trigonometric Functions

2009 ◽  
Author(s):  
Srikanth Tadepalli ◽  
Kristin L. Wood
Keyword(s):  
Experimental Data ◽  
Error Analysis ◽  
Complex Systems ◽  
Drag Coefficient ◽  
Engineering Design ◽  
Numerical Algorithms ◽  
Similarity Analysis ◽  
Trigonometric Functions ◽  
Numerical Approximations ◽  
Prediction Parameters

Similarity methods have been widely employed in engineering design and analysis to model and scale complex systems. The Empirical Similitude Method (ESM) is one such method based on the use of experimental data. Using a variant of the similitude process involving experimental data, we present in this paper, the use of advanced numerical approximations, trigonometric functions in particular to model and predict the performance of design artifacts. Specifically, an airfoil design is modeled, and the values of the drag coefficient are estimated based on the advanced ESM. Intermediate test specimens are used to correlate experimental data to produce the required prediction parameters. Mathematical development and error analysis are also elaborated by delving into continuity and adaptivity features of numerical algorithms.

scholarly journals SOME FORMULAS OF FRACTIONAL TRIGONOMETRIC FUNCTIONS

2021 ◽  
Author(s):  
CHII-HUEI CHII-HUEI
Keyword(s):  
Dirichlet Kernel ◽  
Trigonometric Functions ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Abstract. This paper studies some properties of fractional trigonometric sine and cosine functions and we obtain the fractional Dirichlet kernel. The Mittag-Leffler function plays an important role in this article, and the results we obtained are the generalizations of formulas of the classical sine and cosine functions.

The Trigonometric Functions

1950 ◽  
Vol 43 (5) ◽  
pp. 187-192
Author(s):  
John W. Cell
Keyword(s):  
The Other ◽  
Advanced Mathematics ◽  
Trigonometric Functions ◽  
Points Of View ◽  
Sine And Cosine Functions ◽  
Cosine Functions

In this article we shall indicate many different methods by which the sine and cosine functions may be defined. (From these the other four functions may be obtained by their usual definitions in terms of the sine and cosine functions, viz., cot θ = cos θ/sin θ.) In the course of the discussion we shall consider the trigonometric functions from various points of view and we shall list properties which are not to be found in standard texts on trigonometry but which are found in advanced mathematics. We shall also indicate some general applications which are inherent in these various methods and sources for other applications.

Students' Conceptions of Sine and Cosine Functions When Representing Periodic Motion in a Visual Programming Environment

2018 ◽  
Vol 49 (4) ◽  
pp. 390-423 ◽  
Author(s):  
Anna F. DeJarnette
Keyword(s):  
Prior Knowledge ◽  
Secondary Students ◽  
Periodic Motion ◽  
Visual Programming ◽  
Programming Environment ◽  
Trigonometric Functions ◽  
Sine And Cosine Functions ◽  
Cosine Functions ◽  
Ferris Wheel

In support of efforts to foreground functions as central objects of study in algebra, this study provides evidence of how secondary students use trigonometric functions in contextual tasks. The author examined secondary students' work on a problem involving modeling the periodic motion of a Ferris wheel through the use of a visual programming environment. This study illustrates the range of prior knowledge and resources that students may draw on in their use of trigonometric functions as well as how the goals of students' work inform their reasoning about trigonometric functions.

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