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.

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.

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.

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 TRIGONOMTERY WITHOUT GEOMETRY PART II: GENERALIZATIONS AND APPLICATIONS

2020 ◽  
Vol 72 (1) ◽  
Author(s):  
John Gresham ◽  
Bryant Wyatt ◽  
Jesse Crawford
Keyword(s):  
Analytical Methods ◽  
Geometric Constructions ◽  
Conservation Of Energy ◽  
The Law ◽  
Sine And Cosine Functions ◽  
Cosine Functions

Abstract In a previous paper (Gresham et al. 2019), the properties, theorems, and identities of the sine and cosine functions were developed using only analytical methods and without geometric constructions. We follow those results and use them to develop generalizations of the key theorems of trigonometry, again using purely analytical methods. We conclude with a connection to the Law of Conservation of Energy in physics.

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.

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.

Some Mellin-type Definite Integrals Involving Sine and Cosine Functions

2019 ◽  
Vol 10 (1) ◽  
pp. 222-237
Author(s):  
M. I. Qureshi ◽  
Kaleem A. Quraishi ◽  
Dilshad Ahamad
Keyword(s):  
Definite Integrals ◽  
Sine And Cosine Functions ◽  
Cosine Functions

SINS Installation Error Calibration Based on Multi-Position Combinations

2011 ◽  
Vol 383-390 ◽  
pp. 4213-4220
Author(s):  
Zhen Huan Wang ◽  
Xi Jun Chen ◽  
Qing Shuang Zeng
Keyword(s):  
Theoretical Analysis ◽  
Least Squares ◽  
Rotation Rate ◽  
Scale Factor ◽  
New Method ◽  
Earth’S Rotation ◽  
Earth's Rotation ◽  
Error Calibration ◽  
Sine And Cosine Functions ◽  
Cosine Functions

A new method is proposed to calibrate the installation errors of SINS. According to the method, the installation errors of the gyro and accelerometer can be calibrated simultaneously, which not depend on latitude, gravity, scale factor and earth's rotation rate. By the multi-position combinations, the installation errors of the gyro and accelerometer are modulated into the sine and cosine functions, which can be identified respectively based on the least squares. In order to verify the correctness of the theoretical analysis, the SINS is experimented by a three-axis turntable, and the installation errors of the gyro and accelerometer are identified respectively according to the proposed method. After the compensation of the installation error, the accuracy of the SINS is improved significantly.

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