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Journal articles on the topic 'Trigonometric circle'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

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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2

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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3

Anand, M. Clement Joe, and Janani Bharatraj. "Gaussian Qualitative Trigonometric Functions in a Fuzzy Circle." Advances in Fuzzy Systems 2018 (June 3, 2018): 1–9. http://dx.doi.org/10.1155/2018/8623465.

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We build a bridge between qualitative representation and quantitative representation using fuzzy qualitative trigonometry. A unit circle obtained from fuzzy qualitative representation replaces the quantitative unit circle. Namely, we have developed the concept of a qualitative unit circle from the view of fuzzy theory using Gaussian membership functions, which play a key role in shaping the fuzzy circle and help in obtaining sharper boundaries. We have also developed the trigonometric identities based on qualitative representation by defining trigonometric functions qualitatively and applied t

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Nurbayeva, D. M., Zh M. Nurmukhamedova, S. Yeraliyev, and B. M. Kossanov. "ABOUT DEVELOPMENT OF STUDENTS ' THINKING WHEN SOLVING TRIGONOMETRIC EQUATIONS AND INEQUALITIES IN THE SCHOOL ALGEBRA COURSE." BULLETIN Series of Physics & Mathematical Sciences 69, no. 1 (2020): 138–43. http://dx.doi.org/10.51889/2020-1.1728-7901.23.

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The article deals with solutions of trigonometric inequalities using the unit circle. Specific examples show its application for all trigonometric functions, namely sinus, cosine, tangent and cotangent. An explanation of how to correctly define the period for solving inequalities is also provided. Before analyzing the solution to trigonometric inequalities, the authors present the solution of trigonometric equations according to the formula, but his roots are depicted on the unit circle, where detailed explanation of the record of solutions of this equation. The pictures in the article demonst

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Coghetto, Roland. "Some Facts about Trigonometry and Euclidean Geometry." Formalized Mathematics 22, no. 4 (2014): 313–19. http://dx.doi.org/10.2478/forma-2014-0031.

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Summary We calculate the values of the trigonometric functions for angles: [XXX] , by [16]. After defining some trigonometric identities, we demonstrate conventional trigonometric formulas in the triangle, and the geometric property, by [14], of the triangle inscribed in a semicircle, by the proposition 3.31 in [15]. Then we define the diameter of the circumscribed circle of a triangle using the definition of the area of a triangle and prove some identities of a triangle [9]. We conclude by indicating that the diameter of a circle is twice the length of the radius

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6

Kamber Hamzić, Dina, Mirsad Trumić, and Ismar Hadžalić. "Construction and analysis of test in triangle and circle trigonometry." International Electronic Journal of Mathematics Education 20, no. 1 (2025): em0805. https://doi.org/10.29333/iejme/15734.

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Trigonometry is an important part of secondary school mathematics, but it is usually challenging for students to understand and learn. Since trigonometry is learned and used at a university level in many fields, like physics or geodesy, it is important to have an insight into students’ trigonometry knowledge before the beginning of the university courses. This research aimed to develop a test in triangle and circle trigonometry, which can be used to test students’ prior knowledge of basic trigonometric concepts. A test with multiple-choice questions was developed based on content and learning

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Hašková, V., and M. Lipták. "Processing of calibration measurements on EZB-3." Slovak Journal of Civil Engineering 19, no. 4 (2011): 18–23. http://dx.doi.org/10.2478/v10189-011-0019-7.

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Processing of calibration measurements on EZB-3The calibration of horizontal circles results in a set of discrete correction values. These corrections are obtained in specific circle positions chosen by the size of the calibration step. For further use it is necessary to know the correction values for any location on a horizontal circle; therefore, it is necessary to know the function of the continuous corrections of a horizontal circle. This could be achieved from the measured values by several methods. In the article two methods are presented for determining this function through the approxi

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Sh.L.Ermatov. "TRIGONOMETRIC IDENTITIES OF THE QUADRILATERAL." ACADEMIC RESEARCH IN MODERN SCIENCE 2, no. 10 (2023): 72–73. https://doi.org/10.5281/zenodo.7793925.

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It is known that there are many trigonometric identities regarding the Triangle. These mirrors represent the trigonometric relationship between the inner corners of a triangle, the trigonometric relationship between the inner corners of the Triangle and its main elements (surface, perimeter, sides, radius of the inner and outer drawn circle, bisector, median, height).

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9

MITAN, Carmen-Irena, Emerich BARTHA, Petru FILIP, Constantin DRAGHICI, Miron T. CAPROIU, and Robert M. Moriarty. "Manifold inversion on prediction dihedral angle from vicinal coupling constant with 3-sphere approach." Revue Roumaine de Chimie 68, no. 3-4 (2024): 185–91. http://dx.doi.org/10.33224/rrch.2023.68.3-4.08.

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Dihedral angles are predicted from vicinal coupling constant 3JHH[Hz] with 3-sphere approach, sphere or torus trigonometric equations of circle 1, 2 and circle inversion 3-5 for all cis-, trans-ee, trans-aa stereochemistry. The existence of circle inversion was demonstrated with conformational analysis on five and six membered rings. The sign and the stereochemistry result from vicinal coupling constant under trigonometric equations confirmed by algebraic equations, Hopf and Lie algebra theory. 3-Sphere, a hypersphere in 4D enable for all stereochemistry calculation dihedral angles under magne

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10

Barrera, Azael. "Unit Circles and Inverse Trigonometric Functions." Mathematics Teacher 108, no. 2 (2014): 114–19. http://dx.doi.org/10.5951/mathteacher.108.2.0114.

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11

MELO, WELINGTON DE, PEDRO A. S. SALOMÃO, and EDSON VARGAS. "A full family of multimodal maps on the circle." Ergodic Theory and Dynamical Systems 31, no. 5 (2010): 1325–44. http://dx.doi.org/10.1017/s0143385710000386.

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12

Koch, Hans. "On trigonometric skew-products over irrational circle-rotations." Discrete & Continuous Dynamical Systems 41, no. 11 (2021): 5455. http://dx.doi.org/10.3934/dcds.2021084.

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<p style='text-indent:20px;'>We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that an

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13

Jelitto, Hans. "Generalized Trigonometric Power Sums Covering the Full Circle." Journal of Applied Mathematics and Physics 10, no. 02 (2022): 405–14. http://dx.doi.org/10.4236/jamp.2022.102031.

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14

BISLIMI, Agron. "Continuous Power Flow Analysis in the Trigonometric Circle." PRZEGLĄD ELEKTROTECHNICZNY 1, no. 9 (2024): 133–39. http://dx.doi.org/10.15199/48.2024.09.24.

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15

Sampaio, Junior Cloves Rocha. "Methodology of the isometric polar Cartesian trigonometric circle in the GeoGebra software demonstrating the rationality of the constant π". Núcleo do Conhecimento 05, № 08 (2023): 61–109. https://doi.org/10.32749/nucleodoconhecimento.com.br/mathematical-olympiads/trigonometric-circle.

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The article will present mathematical models demonstrating calculations of the relationships and proportions of the infinite periodic trigonometric circle using the GeoGebra software in the Cartesian, isometric, and polar planes. It will consider the first quadrant of the trigonometric cycle for the calculations of the trigonometric identities functions and apply the same reasoning calculations for the other quadrants. The infinite relationships and proportions between the circumferences’ perimeters and their diameters, the angles (radian arcs) of the circumferences, and square roots. Th

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Lavenda, B. H. "Geometric Entropies of Mixing (EOM)." Open Systems & Information Dynamics 13, no. 01 (2006): 91–101. http://dx.doi.org/10.1007/s11080-006-7270-9.

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Trigonometric and trigonometric-algebraic entropies are introduced and are given an axiomatic characterization. Regularity increases the entropy and the maximal entropy is shown to result when a regular n-gon is inscribed in a circle. A regular n-gon circumscribing a circle gives the largest entropy reduction, or the smallest change in entropy from the state of maximum entropy, which occurs in the asymptotic infinite n-limit. The EOM are shown to correspond to minimum perimeter and maximum area in the theory of convex bodies, and can be used in the prediction of new inequalities for convex set

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17

MOUSSA, ALI. "THE TRIGONOMETRIC FUNCTIONS, AS THEY WERE IN THE ARABIC-ISLAMIC CIVILIZATION." Arabic Sciences and Philosophy 20, no. 1 (2010): 93–104. http://dx.doi.org/10.1017/s0957423909990099.

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AbstractIn the Greek/Indian period, it is noticeable that different radii were used in connection with the chord. This manner continued in the Indian period with the sine, i.e. different sine tables existed. But throughout the Arabic-Islamic period, there was stability in the radius (for the sine). At the time of al-Battānī new terms were introduced, not as functions of angles but as lengths, and again different tables for the same term. Here these terms were not bounded to the circle, and the term miqyās r (measure), which was variable, was used to express “the radius” related to these terms.

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Ларин, С. В., and С. В. Шуманский. "The use of virtual tools of the GeoGebra environment in school trigonometry." Математический вестник Вятского государственного университета, no. 3(26) (April 17, 2023): 49–53. http://dx.doi.org/10.25730/vsu.0536.22.026.

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Анимационные возможности компьютерных сред пополняют арсенал средств построения, добавляя к классическим циркулю и линейке инструменты нового типа, реализуемые на компьютерном экране. Благодаря им в статье моделируется процесс наматывания числовой прямой на единичную окружность, что превращает ее в числовую окружность, моделируется распрямление единичной окружности в отрезок. Кроме того, эти новые виртуальные инструменты положены в основу моделирования непрерывного вычерчивания графиков основных тригонометрических функций и им обратных функций. Это вносит наглядность в преподавание тригонометр

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19

Chen, Xianghong, and Hans Volkmer. "On transfer operators on the circle with trigonometric weights." Journal of Fractal Geometry 5, no. 4 (2018): 351–86. http://dx.doi.org/10.4171/jfg/64.

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20

Wu, Mengjie. "Fast and Accurate Circle Detection Based on Trigonometric Functions." Journal of Information and Computational Science 12, no. 10 (2015): 3863–71. http://dx.doi.org/10.12733/jics20106137.

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21

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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ANAGNOSTOPOULOS, K. N., M. J. BOWICK, and N. ISHIBASHI. "AN OPERATOR FORMALISM FOR UNITARY MATRIX MODELS." Modern Physics Letters A 06, no. 29 (1991): 2727–39. http://dx.doi.org/10.1142/s0217732391003183.

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We analyze the double scaling limit of unitary matrix models in terms of trigonometric orthogonal polynomials on the circle. In particular we find a compact formulation of the string equation at the kth multicritical point in terms of pseudodifferential operators and a corresponding action principle. We also relate this approach to the mKdV hierarchy which appears in the analysis in terms of conventional orthogonal polynomials on the circle.

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23

Sloan, Ian H. "Polynomial approximation on spheres - generalizing de la Vallée-Poussin." Computational Methods in Applied Mathematics 11, no. 4 (2011): 540–52. http://dx.doi.org/10.2478/cmam-2011-0029.

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AbstractFor trigonometric polynomial approximation on a circle, the century-old de la Vallée-Poussin construction has attractive features: it exhibits uniform convergence for all continuous functions as the degree of the trigonometric polynomial goes to infinity, yet it also has arbitrarily fast convergence for sufficiently smooth functions. This paper presents an explicit generalization of the de la Vallée-Poussin construction to higher dimensional spheres S^d ≤ R^{d+1}. The generalization replaces the C^∞ filter introduced by Rustamov by a piecewise polynomial of minimal degree. For the case

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24

Gutmann, Timothy. "A Direct Approach to Computing the Sine or Cosine of the Sum of Two Angles." Mathematics Teacher 96, no. 5 (2003): 314–18. http://dx.doi.org/10.5951/mt.96.5.0314.

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When I teach precalculus, I struggle to decide how much to emphasize different trigonometric identities. Knowing them can help students in subsequent courses, and working with them exposes students to richer applications of algebra than solving one- or two-variable polynomial equations. Yet I often find that the derivations of trigonometric identities are out of place in my classroom, where the focus is on understanding functions as tools for modeling realworld phenomena. There, sine and cosine values are tied directly to measurements of segments within the unit circle.

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Peherstorfer, Franz. "Positive trigonometric quadrature formulas and quadrature on the unit circle." Mathematics of Computation 80, no. 275 (2011): 1685–701. http://dx.doi.org/10.1090/s0025-5718-2011-02414-2.

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26

Khalfallah, Ali Cherif, Hadj Said Naima, and Ali Pacha Adda. "A new chaotic encrypted image based on the trigonometric circle." Information Security Journal: A Global Perspective 29, no. 6 (2020): 297–309. http://dx.doi.org/10.1080/19393555.2020.1767831.

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27

Vernitski, Alexei. "A blind spot in undergraduate mathematics: The circular definition of the length of the circle, and how it can be turned into an enlightening example." MSOR Connections 20, no. 3 (2022): 85–90. http://dx.doi.org/10.21100/msor.v20i3.1300.

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We highlight the fact that in undergraduate calculus, the number pi is defined via the length of the circle, the length of the circle is defined as a certain value of an inverse trigonometric function, and this value is defined via pi, thus forming a circular definition. We present a way in which this error can be rectified. We explain that this error is instructive and can be used as an enlightening topic for discussing different approaches to mathematics with undergraduate students.

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Rababah, Abedallah M. "The best quintic Chebyshev approximation of circular arcs of order ten." International Journal of Electrical and Computer Engineering (IJECE) 9, no. 5 (2019): 3779. http://dx.doi.org/10.11591/ijece.v9i5.pp3779-3785.

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<p>Mathematically, circles are represented by trigonometric parametric equations and implicit equations. Both forms are not proper for computer applications and CAD systems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is of degree $10$ rather than $6$; the Chebyshev error function equioscillates $11$ times rather than $7$; the approximation order is $10$ rather than $6$. The method approximates more than the full circle with Chebyshev uniform error of $1/2^{9}$. The examples show the competence an

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Abedallah, Rababah. "The best quintic chebyshev approximation of circular arcs of order ten." International Journal of Electrical and Computer Engineering (IJECE) 9, no. 5 (2019): 3779–85. https://doi.org/10.11591/ijece.v9i5.pp3779-3785.

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Mathematically, circles are represented by trigonometric parametric equations and im- plicit equations. Both forms are not proper for computer applications and CAD sys- tems. In this paper, a quintic polynomial approximation for a circular arc is presented. This approximation is set so that the error function is of degree 10 rather than 6; the Chebyshev error function equioscillates 11 times rather than 7; the approximation order is 10 rather than 6. The method approximates more than the full circle with Chebyshev uniform error of 1/29. The examples show the competence and simplicity of the pr

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Al-ossmi, Laith H. M. "An elementary treatise on elliptic functions as trigonometry." Alifmatika: Jurnal Pendidikan dan Pembelajaran Matematika 5, no. 1 (2023): 1–20. http://dx.doi.org/10.35316/alifmatika.2023.v5i1.1-20.

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This article concerns the examination of trigonometric identities from an elliptic perspective. The treatment of elliptic functions presented herein adheres to a structure analogous to the traditional exposition of trigonometric functions, with the exception that an ellipse replaces the unit circle. The degree of similarity between the elliptic functions and their trigonometric counterparts is moderated by the periodicity of the so-called El- functions. These identities not only establish the values of the functions, but also establish a correlation between their ratios and the major and minor

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Berriochoa, E., A. Cachafeiro, and J. M. García Amor. "About Nodal Systems for Lagrange Interpolation on the Circle." Journal of Applied Mathematics 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/421340.

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We study the convergence of the Laurent polynomials of Lagrange interpolation on the unit circle for continuous functions satisfying a condition about their modulus of continuity. The novelty of the result is that now the nodal systems are more general than those constituted by thenroots of complex unimodular numbers and the class of functions is different from the usually studied. Moreover, some consequences for the Lagrange interpolation on[-1,1]and the Lagrange trigonometric interpolation are obtained.

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KOVALEV, LEONID V., and XUERUI YANG. "ALGEBRAIC STRUCTURE OF THE RANGE OF A TRIGONOMETRIC POLYNOMIAL." Bulletin of the Australian Mathematical Society 102, no. 2 (2020): 251–60. http://dx.doi.org/10.1017/s0004972719001229.

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The range of a trigonometric polynomial with complex coefficients can be interpreted as the image of the unit circle under a Laurent polynomial. We show that this range is contained in a real algebraic subset of the complex plane. Although the containment may be proper, the difference between the two sets is finite, except for polynomials with a certain symmetry.

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Weisbart, David. "Modernizing Archimedes’ Construction of π". Mathematics 8, № 12 (2020): 2204. http://dx.doi.org/10.3390/math8122204.

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In his famous work, “Measurement of a Circle,” Archimedes described a procedure for measuring both the circumference of a circle and the area it bounds. Implicit in his work is the idea that his procedure defines these quantities. Modern approaches for defining π eschew his method and instead use arguments that are easier to justify, but they involve ideas that are not elementary. This paper makes Archimedes’ measurement procedure rigorous from a modern perspective. In so doing, it brings a rigorous and geometric treatment of the differential properties of the trigonometric functions into the

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Du, Mao Lin, Guo Liang Chen, and Kui Ying Pu. "Simplification of the Parametric Transformation Equations for Analysis of Plane Stress." Advanced Materials Research 591-593 (November 2012): 2499–503. http://dx.doi.org/10.4028/www.scientific.net/amr.591-593.2499.

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For the analysis of plane stress, there are two methods, transformation of equation and Mohr’s circle. With the former method, the two principle angles are found by taking derivative to determine the extreme value of stresses. However, it is unknown which one of the two principle angles corresponding to the maximum stress. Moreover, the relationship between the two methods is inexplicit, as the angle of the rotation of the element is eliminated to derive the equation of Mohr’s circle. Using trigonometric relations, considering two systems of sign conventions for shear stresses, simplified para

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Borrego-Morell, Jorge A., Cleonice F. Bracciali, and Alagacone Sri Ranga. "On an Energy-Dependent Quantum System with Solutions in Terms of a Class of Hypergeometric Para-Orthogonal Polynomials on the Unit Circle." Mathematics 8, no. 7 (2020): 1161. http://dx.doi.org/10.3390/math8071161.

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We study an energy-dependent potential related to the Rosen–Morse potential. We give in closed-form the expression of a system of eigenfunctions of the Schrödinger operator in terms of a class of functions associated to a family of hypergeometric para-orthogonal polynomials on the unit circle. We also present modified relations of orthogonality and an asymptotic formula. Consequently, bound state solutions can be obtained for some values of the parameters that define the model. As a particular case, we obtain the symmetric trigonometric Rosen–Morse potential for which there exists an orthogona

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El-Mongy, Sayed. "New Theorem and Formula for Circle Arc Length Calculations with Trigonometric Approach Application in Astrophysics." JOURNAL OF ADVANCES IN PHYSICS 18 (November 20, 2020): 158–63. http://dx.doi.org/10.24297/jap.v18i.8914.

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The circle and sphere have been studied since the ancient Egyptians and Greeks before the Common Era (BCE). The recent scientific renaissance has also used them in different fields. It is also mentioned in the Prophet Mohamed`s Holy Quran. This article introduces a new Theorem (S. El-Mongy`s Theorem) as an empirical formula to correlate the constant (e) with circle and sphere. It states that “the arc length is correlated as a direct function in {(e π r sA)}, whatever the central angle (ϴ) and radius (r). The factor sA is (ϴ/10ϕ). The formula can also be written as; AL = {(0.0174533185 r ϴ)}. W

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Shpilev, P. V. "L-optimal designs in the trigonometric regression model on the full circle." Vestnik St. Petersburg University: Mathematics 40, no. 2 (2007): 158–68. http://dx.doi.org/10.3103/s1063454107020112.

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Fakharzadeh, J. A., and F. N. Jafarpoor. "A Solution Method for an Optimal Controlled Vibrating Circle Shell by Measure and Classical Trajectory." Applied Mechanics and Materials 52-54 (March 2011): 1855–60. http://dx.doi.org/10.4028/www.scientific.net/amm.52-54.1855.

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The mean idea of this paper is to present a new combinatorial solution technique for the controlled vibrating circle shell systems. Based on the classical results of the wave equations on circle domains, the trajectory is considered as a finite trigonometric series with unknown coefficients in polar coordinates. Then, the problem is transferred to one in which its unknowns are a positive Radon measure and some positive coefficients. Extending the underlying space helps us to prove the existence of the solution. By using the density properties and some approximation schemes, the problem is defo

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Bajwa, Sewa Singh. "Evolution and Significance of Sine Function in Ancient Indian Mathematics." Journal of Educational Research and Policies 6, no. 11 (2024): 35–38. https://doi.org/10.53469/jerp.2024.06(11).08.

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This research article delves into the historical and mathematical intricacies of the construction of sine functions, particularly focusing on ancient Indian mathematical concepts and their evolution. It begins by highlighting the significance of sine (jya) as a fundamental trigonometric function in Indian mathematics, alongside its synonyms in Sanskrit. The historical journey of the term “sine” is traced back to ancient Indian mathematical texts, which were later translated into Arabic and then into Latin, ultimately leading to the adoption of the term “sine” in modern mathematics. The article

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Reid, Mark J. "Galactic structure from trigonometric parallaxes of star-forming regions." Proceedings of the International Astronomical Union 8, S289 (2012): 188–93. http://dx.doi.org/10.1017/s1743921312021369.

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AbstractRecently, astrometric accuracy approaching ~ 10 μas has become routinely possible with Very Long Baseline Interferometry. Since, unlike at optical wavelengths, interstellar dust is transparent at radio wavelengths, parallaxes and proper motions can now be measured for massive young stars (with maser emission) across the Galaxy, enabling direct measurements of the spiral structure of the Milky Way. Fitting the full 3D position and velocity vectors to a simple model of the Galaxy yields extremely accurate values for its fundamental parameters, including the distance to the Galactic Cente

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Гончарова, З. Г., Т. Ю. Дёмина, Е. В. Неискашова, and В. В. Демин. "One of the methods of root selection in solving trigonometric equations." Management of Education, no. 3 (May 30, 2021): 159–68. http://dx.doi.org/10.25726/r7854-2774-0932-h.

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При подготовке учащихся 10-11 классов к профильному ЕГЭ по математике возникают трудности при отборе корней тригонометрического уравнения, которые принадлежат заданному промежутку. Существует несколько методов отбора корней, но идеального не существует – у каждого из этих методов есть свои слабые стороны. Мы хотим предложить метод, который, на наш взгляд, позволяет учащимся более успешно производить отбор корней в тригонометрических уравнениях. В школьном курсе математики для отбора корней чаще всего используются тригонометрический круг или отбор корней с помощью двойного неравенства, определя

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Maknun, C. L., R. Rosjanuardi, and A. Jupri. "From ratios of right triangle to unit circle: an introduction to trigonometric functions." Journal of Physics: Conference Series 1157 (February 2019): 022124. http://dx.doi.org/10.1088/1742-6596/1157/2/022124.

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Chang, Fu-Chuen, and Chin-Han Li. "D-optimal designs for combined polynomial and trigonometric regression on a partial circle." Journal of Statistical Planning and Inference 143, no. 7 (2013): 1186–94. http://dx.doi.org/10.1016/j.jspi.2013.01.013.

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Dana-Picard, Thierry, and Tomás Recio. "Dynamic construction of a family of octic curves as geometric loci." AIMS Mathematics 8, no. 8 (2023): 19461–76. http://dx.doi.org/10.3934/math.2023993.

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<abstract><p>We explore the construction of curves of degree 8 (octics) appearing as geometric loci of points defined by moving points on an ellipse and its director circle. To achieve this goal we develop different computer algebra methods, dealing with trigonometric or with rational parametric representations, as well as through implicit polynomial equations, of the given curves. Finally, we highlight the involved mathematical or computational issues arising when reflecting on the outputs obtained in each case.</p></abstract>

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Amortila, Valentin, Elena Mereuta, Silvia Veresiu, Madalina Rus, and Costel Humelnicu. "Positioning study of driver's hands in certain areas of the steering wheel." MATEC Web of Conferences 178 (2018): 06014. http://dx.doi.org/10.1051/matecconf/201817806014.

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The aim of this paper is to analyze driver's hands coordinates on the steering wheel for an optimal and safe driving experience. A good coordination of the driver's action on the controls is the result of a comfortable position that leads to an optimal reaction while driving. The presented study implies using a thermal imaging camera for analysing palms temperature changes in the contact area with the steering wheel. The resulting data shows that the optimal driving position of drivers' hands is 0° and 180° associating the steering wheel with and trigonometric circle.

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Stavek, Jirí. "On the Trigonometric Descriptions of Colors." Applied Physics Research 8, no. 3 (2016): 5. http://dx.doi.org/10.5539/apr.v8n3p5.

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<p class="1Body">An attempt is presented for the description of the spectral colors using the standard trigonometric tools in order to extract more information about photons. We have arranged the spectral colors on an arc of the circle with the radius R = 1 and the central angle θ = π/3 when we have defined cos (θ) = λ<sub>380</sub>/λ<sub>760</sub> = 0.5. Several trigonometric operations were applied in order to find the gravity centers for the scotopic, photopic, and mesopic visions. The concept of the center of gravity of colors introduced Isaac Newton. We have

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Maknun, Churun Lu’lu’il, Rizky Rosjanuardi, and Al Jupri. "Didactical Design on Drawing and Analysing Trigonometric Functions Graph through a Unit Circle Approach." International Electronic Journal of Mathematics Education 15, no. 3 (2020): em0614. http://dx.doi.org/10.29333/iejme/9275.

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Altman, Renana, and Ivy Kidron. "Constructing knowledge about the trigonometric functions and their geometric meaning on the unit circle." International Journal of Mathematical Education in Science and Technology 47, no. 7 (2016): 1048–60. http://dx.doi.org/10.1080/0020739x.2016.1189005.

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Chang, Fu-Chuen, Lorens Imhof, and Yi-Ying Sun. "Exact $$D$$ -optimal designs for first-order trigonometric regression models on a partial circle." Metrika 76, no. 6 (2012): 857–72. http://dx.doi.org/10.1007/s00184-012-0420-x.

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Lubinsky, D. S. "ON MEAN CONVERGENCE OF TRIGONOMETRIC INTERPOLANTS, AND THEIR UNIT CIRCLE ANALOGUES, FOR GENERAL ARRAYS." Analysis 22, no. 1 (2002): 97–108. http://dx.doi.org/10.1524/anly.2002.22.1.97.

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