Trigonometry

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Temur Z Kalanov - One of the best experts on this subject based on the ideXlab platform.

  • the critical analysis of the mathematical formalism of theoretical physics iv the foundations of Trigonometry
    viXra, 2014
    Co-Authors: Temur Z Kalanov
    Abstract:

    Analysis of the foundations of standard Trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of Trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of Trigonometry: standard Trigonometry is a false theory.

  • On the System Analysis of the Foundations of Trigonometry
    Pure and Applied Mathematics Journal, 2014
    Co-Authors: Temur Z Kalanov
    Abstract:

    Analysis of the foundations of standard Trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of Trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of Trigonometry: standard Trigonometry is a false theory.

  • critical analysis of the mathematical formalism of theoretical physics iv Trigonometry
    viXra, 2013
    Co-Authors: Temur Z Kalanov
    Abstract:

    Analysis of the foundations of standard Trigonometry is proposed. The unity of formal logic and of rational dialectics is methodological basis of the analysis. It is shown that the foundations of Trigonometry contradict to the principles of system approach and contain formal-logical errors. The principal logical error is that the definitions of trigonometric functions represent quantitative relationships between the different qualities: between qualitative determinacy of angle and qualitative determinacy of rectilinear segments (legs) in rectangular triangle. These relationships do not satisfy the standard definition of mathematical function because there are no mathematical operations that should be carry out on qualitative determinacy of angle to obtain qualitative determinacy of legs. Therefore, the left-hand and right-hand sides of the standard mathematical definitions have no the identical sense. The logical errors determine the essence of Trigonometry: standard Trigonometry is a false theory.

Karl Gustafson - One of the best experts on this subject based on the ideXlab platform.

  • A Trigonometry of Quantum States
    2010
    Co-Authors: Karl Gustafson
    Abstract:

    Recently the geometry of quantum states has been under considerable development. Every good geometry deserves, if possible, an accompanying Trigonometry. I will here introduce such a Trigonometry to accompany the geometry of quantum states.

  • the Trigonometry of matrix statistics
    International Statistical Review, 2006
    Co-Authors: Karl Gustafson
    Abstract:

    Summary A matrix Trigonometry developed chiefly by this author during the past 40 years has interesting applications to certain situations in statistics. The key conceptual entity in this matrix Trigonometry is the matrix (maximal) turning angle. Associated entities (originally so-named by this author) are the matrix antieigenvalues and corresponding antieigenvectors upon which the matrix obtains its critical turning angles. Because this Trigonometry is the natural one for linear operators and matrices, it also is the natural one for matrix statistics. Resume Une trigonometrie matricielle developpee principalement par cet auteur dans les 40 dernieres annees a des applications interessantes a certaines situations en statistique. L'entite conceptuelle cle dans cette trigonometrie matricielle est l'angle de rotation (maximal). Les entites associees (ainsi nommees a l'origine par cet auteur) sont les anti valeurs propres et les anti vecteurs propres sur lesquels la matrice obtient ses angles de rotation critiques. Comme cette trigonometrie est naturelle pour les operateurs lineaires et les matrices, elle est aussi naturelle pour les statistiques matricielles.

  • an extended operator Trigonometry
    Linear Algebra and its Applications, 2000
    Co-Authors: Karl Gustafson
    Abstract:

    The operator Trigonometry of symmetric positive definite (SPD) matrices is extended to arbitrary invertible matrices A and to arbitrary invertible bounded operators A on a Hilbert space. Some background and motivation for these results is provided.

  • Operator Trigonometry of Linear Systems
    IFAC Proceedings Volumes, 1998
    Co-Authors: Karl Gustafson
    Abstract:

    Abstract Operator Trigonometry, which originated in problems of functional analysis in the late 1960's, is applied to iterative methods for the solution of linear systems Ax = b. The operator Trigonometry provides a new theory for the understanding and analysis of such methods. For example, the famous Kantorovich error bound for gradient algorithms is now seen to be inherently trigonometric. The basic Richardson scheme also turns out to be fundamentally trigonometric. Stability of control is also seen trigonometrically.

Mariano Santander - One of the best experts on this subject based on the ideXlab platform.

  • Trigonometry of spacetimes a new self dual approach to a curvature signature in dependent Trigonometry
    Journal of Physics A, 2000
    Co-Authors: Francisco J. Herranz, Ramón Ortega, Mariano Santander
    Abstract:

    A new method to obtain Trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method could be described as `curvature/signature (in)dependent Trigonometry' and encapsulates Trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an `absolute Trigonometry', and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic spacetimes; therefore a complete discussion of Trigonometry in the six de Sitter, Minkowskian, Newton-Hooke and Galilean spacetimes follow as particular instances of the general approach. Distinctive traits of the method are `universality' and `self-duality': every equation is meaningful for the nine spaces at once, and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a complete study of Trigonometry and, in fact, any equation previously known for the three classical (Riemannian) spaces also has a version for the remaining six `spacetimes'; in most cases these equations are new.

  • Trigonometry of spacetimes: a new self-dual approach to a curvature/signature (in)dependent Trigonometry
    Journal of Physics A: Mathematical and General, 2000
    Co-Authors: Francisco J. Herranz, Ramón Ortega, Mariano Santander
    Abstract:

    A new method to obtain Trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method could be described as `curvature/signature (in)dependent Trigonometry' and encapsulates Trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an `absolute Trigonometry', and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic spacetimes; therefore a complete discussion of Trigonometry in the six de Sitter, Minkowskian, Newton-Hooke and Galilean spacetimes follow as particular instances of the general approach. Distinctive traits of the method are `universality' and `self-duality': every equation is meaningful for the nine spaces at once, and displays invariance explicitly under a duality transformation relating the nine spaces amongst themselves. These basic structural properties allow a complete study of Trigonometry and, in fact, any equation previously known for the three classical (Riemannian) spaces also has a version for the remaining six `spacetimes'; in most cases these equations are new.

  • Trigonometry of spacetimes a new self dual approach to a curvature signature in dependent Trigonometry
    arXiv: Mathematical Physics, 1999
    Co-Authors: Francisco J. Herranz, Ramón Ortega, Mariano Santander
    Abstract:

    A new method to obtain Trigonometry for the real spaces of constant curvature and metric of any (even degenerate) signature is presented. The method encapsulates Trigonometry for all these spaces into a single basic trigonometric group equation. This brings to its logical end the idea of an absolute Trigonometry, and provides equations which hold true for the nine two-dimensional spaces of constant curvature and any signature. This family of spaces includes both relativistic and non-relativistic homogeneous spacetimes; therefore a complete discussion of Trigonometry in the six de Sitter, minkowskian, Newton--Hooke and galilean spacetimes follow as particular instances of the general approach. Any equation previously known for the three classical riemannian spaces also has a version for the remaining six spacetimes; in most cases these equations are new. Distinctive traits of the method are universality and self-duality: every equation is meaningful for the nine spaces at once, and displays explicitly invariance under a duality transformation relating the nine spaces. The derivation of the single basic trigonometric equation at group level, its translation to a set of equations (cosine, sine and dual cosine laws) and the natural apparition of angular and lateral excesses, area and coarea are explicitly discussed in detail. The exposition also aims to introduce the main ideas of this direct group theoretical way to Trigonometry, and may well provide a path to systematically study Trigonometry for any homogeneous symmetric space.

P. Srinivas - One of the best experts on this subject based on the ideXlab platform.

  • KBCS - The Trigonometry tutor
    Knowledge Based Computer Systems, 1
    Co-Authors: Parvati Rajan, Pramad Patil, K. S. R. Anjaneyulu, P. Srinivas
    Abstract:

    This paper describes a tutor which teaches students trigonometric problem solving. We first discuss approaches others have adopted for teaching mathematics and comment on their usefulness in Trigonometry. We later present a task analysis of Trigonometry, based on an experiment we conducted. We then present our approach to teach different categories of problems in Trigonometry.

Jesse L. M. Wilkins - One of the best experts on this subject based on the ideXlab platform.

  • Mathematics Coursework Regulates Growth in Mathematics Achievement
    Journal for Research in Mathematics Education, 2007
    Co-Authors: Jesse L. M. Wilkins
    Abstract:

    Using data from the Longitudinal Study of American Youth (LSAY), we examined the extent to which students mathematics coursework regulates (influences) the rate of growth in mathematics achievement during middle and high school. Graphical analysis showed that students who started middle school with higher achievement took individual mathematics courses earlier than those with lower achievement. Immediate improvement in mathematics achievement was observed right after taking particular mathematics courses (regular mathematics, prealgebra, algebra I, Trigonometry, and calculus). Statistical analysis showed that all mathematics courses added significantly to growth in mathematics achievement, although this added growth varied significantly across students. Regular mathematics courses demonstrated the least regulating power, whereas advanced mathematics courses (Trigonometry, precalculus, and calculus) demonstrated the greatest regulating power. Regular mathematics, prealgebra, algebra I, geometry, and Trigonometry were important to growth in mathematics achievement even after adjusting for more advanced courses taken later in the sequence of students' mathematics coursework

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