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 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.

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 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

Sin is Dead, Long Live Sin!

2017 ◽  
Author(s):  
Kate Kirkpatrick
Keyword(s):  
Religious Commitment ◽  
The Other ◽  
Realist Approach ◽  
Points Of View ◽  
Cultural Amnesia

Chapter 9 offers two concluding ‘provocations’: one on wretchedness without God, the other on wretchedness with God. The first brings Sartre into dialogue with Marilyn McCord Adams’s work ‘God because of Evil’, arguing that Sartre’s account lends credence to her view that optimism is not warranted if one takes a robust realist approach to evil. Read as a phenomenologist of fallenness, Sartre may serve the apologetic purpose of making options ‘live’, in William James’s language; or, to use the phrase of Stephen Mulhall, to ‘hold open the possibility of taking religious points of view seriously’. The second provocation—on the question of wretchedness with God—suggests that Sartre can be read ‘for edification’ to help us see our failures in love. The book concludes that reading Sartre in this light can help redress ‘damaging cultural amnesia’ about religious commitment, offering an account of sin that cultivates humility, love, and mercy.

Combinatorial principles weaker than Ramsey's Theorem for pairs

2007 ◽  
Vol 72 (1) ◽  
pp. 171-206 ◽  
Author(s):  
Denis R. Hirschfeldt ◽  
Richard A. Shore
Keyword(s):  
Reverse Mathematics ◽  
The Other ◽  
Base System ◽  
Ramsey's Theorem ◽  
Open Questions ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Points Of View ◽  
The One ◽  
Infinite Linear

AbstractWe investigate the complexity of various combinatorial theorems about linear and partial orders, from the points of view of computability theory and reverse mathematics. We focus in particular on the principles ADS (Ascending or Descending Sequence), which states that every infinite linear order has either an infinite descending sequence or an infinite ascending sequence, and CAC (Chain-AntiChain), which states that every infinite partial order has either an infinite chain or an infinite antichain. It is wellknown that Ramsey's Theorem for pairs () splits into a stable version () and a cohesive principle (COH). We show that the same is true of ADS and CAC, and that in their cases the stable versions are strictly weaker than the full ones (which is not known to be the case for and ). We also analyze the relationships between these principles and other systems and principles previously studied by reverse mathematics, such as WKL0, DNR, and BΣ2. We show, for instance, that WKL0 is incomparable with all of the systems we study. We also prove computability-theoretic and conservation results for them. Among these results are a strengthening of the fact, proved by Cholak, Jockusch, and Slaman, that COH is -conservative over the base system RCA0. We also prove that CAC does not imply DNR which, combined with a recent result of Hirschfeldt, Jockusch. Kjos-Hanssen, Lempp, and Slaman, shows that CAC does not imply (and so does not imply ). This answers a question of Cholak, Jockusch, and Slaman.Our proofs suggest that the essential distinction between ADS and CAC on the one hand and on the other is that the colorings needed for our analysis are in some way transitive. We formalize this intuition as the notions of transitive and semitransitive colorings and show that the existence of homogeneous sets for such colorings is equivalent to ADS and CAC, respectively. We finish with several open questions.

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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