Dissertations / Theses: 'Trigonometry functions'

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Jennings, Paul Richard. "Hyperspherical trigonometry, related elliptic functions and integrable systems." Thesis, University of Leeds, 2013. http://etheses.whiterose.ac.uk/6892/.

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The basic formulae of hyperspherical trigonometry in multi-dimensional Euclidean space are developed using multi-dimensional vector products, and their conversion to identities for elliptic functions is shown. The basic addition formulae for functions on the 3-sphere embedded in four-dimensional space are shown to lead to addition formulae for elliptic functions, associated with algebraic curves, which have two distinct moduli. Application of these formulae to the cases of a multi-dimensional Euler top and Double Elliptic Systems are given, providing a connection between the two. A generalisation of the Lattice Potential Kadomtsev-Petviashvili (LPKP) equation is presented, using the method of Direct Linearisation based on an elliptic Cauchy kernel. This yields a (3 + 1)-dimensional lattice system with one of the lattice shifts singled out. The integrability of the lattice system is considered, presenting a Lax representation and soliton solutions. An associated continuous system is also derived, yielding a (3 + 1)- dimensional generalisation of the potential KP equation associated with an elliptic curve.

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Silva, Jander Carlos Silva e. "As novas tecnologias no contexto escolar: uma abordagem sobre aplicações do GeoGebra em trigonometria." Universidade de São Paulo, 2015. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-17122015-104430/.

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Este trabalho apresenta uma abordagem sobre as novas tecnologias no contexto escolar, com vistas para aplicação do GeoGebra em trigonometria. O objetivo é nortear professores da educação básica na preparação de aulas usando o GeoGebra, visando ao enriquecimento do tema trigonometria em sala de aula. As atividades propostas estão divididas em três grupos: trigonometria básica, funções trigonométricas e equações trigonométricas. Cada uma possui um alto nível de detalhamento, com o objetivo de incentivar o uso por professores com pouco ou nenhum conhecimento do software, bem como incentivar atividades que promovam a criação por parte dos alunos. A ideia é que os alunos construam as atividades, aprendendo a utilizar o software, interagindo por meio da movimentação dos objetos, e tirando suas conclusões pertinentes às atividades. De maneira geral, pretende-se contribuir para o desenvolvimento do raciocínio lógico do aluno por meio do ensino de Matemática agregando a utilização de tecnologia, de forma que o aluno não seja somente um expectador, mas sim, participante da construção da própria atividade.
This work presents na approach to new Technologies in the educational context, with a view to applications of the GeoGebra in trigonometry. The goal is to guide teachers of the basic education in preparing lessons using GeoGebra, aiming to enrich trigonometry the in the classroom. The proposed activities are divided into three groups : basic trigonometry, trigonometry functions and trigonometry equations. Each one has a high level of details, in order to encourage the use by teachers with little or no knowledge of the software, and also encourage activities that promote the creation by the students. The idea is that students build the activities, learning how to use the software, interacting by moving objects, and taking their conclusions about the activities. In general, one intends to contribute to the development of logical thinking of students through the teaching of Mathematics adding the use of technology, so that the student is not only a spectator, but, participant of the construction of their own activity.

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Baines, Clare Elizabeth. "Topics in functions with symmetry." Thesis, University of Liverpool, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.343778.

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Malambo, Priestly. "Exploring Zambian Mathematics student teachers' content knowledge of functions and trigonometry for secondary schools." Thesis, University of Pretoria, 2015. http://hdl.handle.net/2263/52943.

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This was a qualitative case study that explored Zambian mathematics student teachers? content knowledge of functions and trigonometry in secondary schools. The students were in their final year of study and had studied advanced university mathematics and had completed Mathematics Education courses. Content knowledge was investigated as Common Content Knowledge (CCK) and Specialised Content Knowledge (SCK). Data was collected in two phases: Phase 1 utilised a mathematics test to gather CCK and SCK data from 22 conveniently chosen University of Zambia student teachers majoring in mathematics. Phase 2 used semi-structured interviews to collect SCK data from a sub-sample of six purposefully selected students. Descriptive statistics and qualitative techniques were used to analyse the test data and content analysis to analyse the interviews. Although the students achieved a mean score of about 52% in the CCK of functions, an item by item analysis suggested that they were not proficient therein. They had a shallow understanding of composite functions, domains and ranges, extreme values, and turning points. They also had a superficial understanding of the definitions of concepts. While the students managed to identify functions, the majority could not coherently explain concepts and justify their reasoning. The students showed a limited understanding of the Cartesian plane representations of functions and the algebraic representation of quadratic functions. The students achieved a mean score of approximately 53% in the CCK of trigonometry, and 68% of the sample achieved scores above 50%. An item by item analysis suggests that most of the students were proficient in CCK. However, the students could not comprehensively explain concepts and justify their reasoning. While most of the students could apply rules and formulas, they could not coherently explain and prove these. Similarly, they could not translate algebraic trigonometric functions to the Cartesian plane. Generally, there seemed to be a disconnection between the students? CCK and SCK of trigonometry. These findings suggested that the study of advanced mathematics does not automatically result in students? comprehensive understanding of school mathematics. While the students had studied advanced UNZA mathematics, it was found out that they had not acquired an in-depth understanding of the functions and trigonometry required at secondary school level.
Thesis (PhD)--University of Pretoria, 2015.
Science, Mathematics and Technology Education
PhD

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Yadollahi, Farsani Leila. "Topics in the calculus of variations : quasiconvexification of distance functions and geometry in the space of matrices." Thesis, University of Sussex, 2017. http://sro.sussex.ac.uk/id/eprint/68825/.

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OLIVEIRA, Carlos André Carneiro de. "Trigonometria: o radiano e as funções seno, cosseno e tangente." Universidade Federal de Campina Grande, 2014. http://dspace.sti.ufcg.edu.br:8080/jspui/handle/riufcg/2169.

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Este trabalho apresenta um estudo sobre o ensino da trigonometria no ensino médio, contemplando as recomendações sobre esse conteúdo encontradas nos Parâmetros Curriculares Nacionais e uma breve análise desses conteúdos em alguns dos livros recomendados pelo Guia de Livros Didáticos de Matemática - PNLP 2012. Destacando a formação do conceito de radiano; a extensão das razões trigonométricas seno, cosseno e tangente definidas no triângulo retângulo para as funções Trigonométricas de domínio real, além das demonstrações geométricas das fórmulas da adição e da subtração de arcos das funções seno, cosseno e tangente. Apresenta, também, uma sequência didática, com atividades contemplando os conteúdos destacados acima. As atividades foram elaboradas tendo como referência a teoria da aprendizagem significativa e adaptadas ao uso do software GeoGebra.
This work presents a study on the teaching of trigonometry in high school, in accordance with the recommendations about this subject found in National Curriculum Guidelines (Parâmetros Curriculares Nacionais) and a analysis of the content of some of the books recommended by the Mathematics Textbook Guide - PNLP 2012. It highlight the formation of the concept of radian; the extension of trigonometric ratios sine, cosine and tangent defined in the triangle for Trigonometric functions of real field, in addition to the geometrical proofs of the formula of addition and subtraction arches functions sine, cosine and tangent. It also presents a didactic sequence, with activities covering the highlighted contents above. The activities were developed with reference to the meaningful learning theory and adapted to the use of GeoGebra software.

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Calderaro, André Bispo. "Análise da possibilidade de inclusão de abordagens alternativas para a função cosseno no ensino médio." Mestrado Profissional em Matemática, 2013. https://ri.ufs.br/handle/riufs/6504.

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This work establishes itself as objective analysis of the process of elaboration of the definitions of the trigonometric functions, particularly the definition of the cosine function. Thus, it is given a guideline regarding the development of a curriculum in which students are not sparked off the concept of trigonometric function based only on the triangle and trigonometric cycle. Therefore, the biggest challenge is to develop arguments for teaching about and investigate further the concepts of trigonometry. Thus, although we like the first chapter a brief explanation about the history of trigonometry, the work focuses on the development of the definition cosine of some forms, namely: the one presented in textbooks as it observes that it is made with support Euler function, setting via power series, the setting based on the exponential function domain in complex and, finally, setting as a solution of an initial value problem, that is, a differential equation that satisfies a given initial condition. From there will be an analysis of these definitions by observing the viability of each to be presented to high school students.
Neste trabalho, se estabelece como objetivo a análise do processo de elaboração das definiçõoes das funçõoes trigonométricas, particularmente, a definição da funçãao cosseno. Desse modo, é dada uma diretriz no que toca a elaboração de um currículo em que os alunos não fiquem ateados ao conceito de função trigonométrica apenas com base no triângulo retângulo e no ciclo trigonométrico. Portanto, o maior desafio é desenvolver argumentos didáticos para tal, bem como investigar mais detalhadamente as concepções da trigonometria. Com isso, apesar de termos como primeiro capítulo uma breve explanação sobre a história da trigonometria, o trabalho tem como foco o desenvolvimento da definição cosseno de algumas formas, sendo elas: a apresentada nos livros didáticos em que se observa que a mesma é feita com suporte da função de Euler, a definição através de séries de potências, a definição com base na função exponencial com domínio nos complexos e, por fim, a definição como solução de um problema de valor inicial, isto é, uma equação diferencial que satisfaz uma determinada condição inicial. A partir daí será feita uma análise dessas definições observando-se a viabilidade de cada uma delas ser apresentada para os alunos do Ensino Médio.

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Souza, Francine Dalavale Tozatto. "Trigonometria no ensino médio e suas aplicações." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55136/tde-26102018-170937/.

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Neste trabalho fazemos um estudo detalhado sobre o tema Trigonometria. A trigonometria é um tema bastante discutido em sala de aula durante o ensino médio. Não apenas apresentamos resultados sobre o tema mas também suas provas e justificativas, assim como exemplos e exercícios com o objetivo de ter um material completo para professores do ensino médio que desejem estudar tais tópicos. Em seguida apresentamos algumas aplicações da Trigonometria que podemos encontrar em nosso dia-a-dia, também aqui o objetivo é apresentar motivação para o estudo deste importante assunto e tão frequente nos vestibulares atualmente. Finalmente, apresentamos uma atividade realizada com meus alunos em sala de aula. Esta dissertação foi desenvolvida como parte dos requisitos necessários para a obtenção do título de mestrado acadêmico junto ao Instituto de Ciências Matemáticas e de Computação (ICMC), da Universidade de São Paulo (USP).
In this dissertation we present a detailed study about Trigonometry. This subject is frequently discussed em classes during High school courses. We do not only present the main results about Trigonometry but also their proofs, as well examples and exercises. Our main objective here is obtain a complete text for high school teachers. We also present some applications of Trigonometry that can be easily find in our life. Here our main objective is to motivate the study of this important subject that appears so frequently in the exams for universities entrance. To conclude, we present an activity realized with high school students. This dissertation was developed as part of the requirements necessary for the obtension of the degree of Mathematics Professional Master at Instituto de Ciências Matemáticas e de Computação da Universidade de São Paulo (ICMC-USP).

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Bruginski, Willian José. "Desenvolvimento de planilhas dinâmicas utilizando o software Geogebra para o estudo de funções trigonométricas." Universidade Tecnológica Federal do Paraná, 2014. http://repositorio.utfpr.edu.br/jspui/handle/1/802.

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Esta dissertação foi desenvolvida com o intuito de criar uma ferramenta para auxiliar no ensino da trigonometria. A ferramenta foi criada com o apoio de recursos tecnológicos e o Geogebra foi o software escolhido para a elaboração deste projeto. Devido a quantidade de recursos que o software dispõe principalmente a possibilidade de trabalhar de forma integrada a geometria com a álgebra, este foi um grande aliado na criação das planilhas dinâmicas. Na sequencia foi desenvolvida a parte teórica das funções trigonométricas com as suas definições, as características, construções dos gráficos e foram apresentadas as contribuições que as planilhas dinâmicas proporcionam neste estudo.
This work was developed with the intention of creating a new tool to assist in teaching trigonometry. The tool has been created with the support of technological resources and Geogebra software has been chosen for the development of this project. Because the amount of resources that the software provides, especially the ability to work seamlessly geometry and algebra, this was a great ally in the creation of dynamic spreadsheets. Following was developed theoretical part of trigonometric functions with their definitions, characteristics, construction of graphs and contributions that dynamic spreadsheets provide this study were presented.

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Dionizio, Fátima Aparecida Queiroz. "CONHECIMENTOS DOCENTES: UMA ANÁLISE DOS DISCURSOS DE PROFESSORES QUE ENSINAM MATEMÁTICA." UNIVERSIDADE ESTADUAL DE PONTA GROSSA, 2013. http://tede2.uepg.br/jspui/handle/prefix/1349.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
This study presents an analysis of pedagogical, curricular and content knowledge that teachers of mathematics in basic education, in Ponta Grossa/PR have on Trigonometry. Empirical data was obtained with the use of a tool containing different kinds of errors made by students in Trigonometry exercises to be analyzed and point out possible intervention by the teachers as well as to present responses to Trigonometry concepts. This data was analyzed based on Shulman’s (1986,2001) theoretical contributions and on the functions and discursive operations according to the Register of Semiotic Representation theory by Raymond Duval (2004, 2009, 2011). The research question was: How is knowledge about Trigonometry characterized and what is the nature of the knowledge related to learning issues and Trigonometry teaching presented by mathematics teachers who work in basic education? And the objectives of the study were: to characterize the knowledge of Trigonometry of mathematics teachers working in basic education and; to reveal the nature of mathematics teachers’ knowledge in relation to Trigonometry learning and teaching issues.The research had a qualitative approach, with descriptive and elucidative characteristics, and the analysis methodological procedures were assisted by Bardin’s (2009) content analysis. The organization of data was carried out with the aid of the software Atlas.ti, through which the subject content knowledge, the pedagogical knowledge of content, and the curricular knowledge were gathered within the discourse expansion function proposed by Duval (2004) for later analysis. The results of this study indicate that teachers’ knowledge that was more evident was the knowledge of the subject content. These results point to the need for teachers to be more attentive to other aspects of knowledge also necessary to their education practice and which seem to have been disregarded.
Este trabalho apresenta uma análise dos conhecimentos pedagógicos, curricular e de conteúdo de professores de matemática da Educação Básica, do município de Ponta Grossa/PR, sobre Trigonometria. Os dados empíricos foram obtidos por meio da aplicação de um instrumento contendo diferentes tipos de erros cometidos pelos alunos em atividades de Trigonometria, para a análise e apontamento de possíveis intervenções pedagógicas pelos professores e também respostas a conceitos de Trigonometria. Esses dados foram analisados tendo por subsídios teóricos as contribuições de Shulman (1986, 2001) e as funções e operações discursivas apresentadas na teoria dos Registros de Representação Semiótica segundo Raymond Duval (2004, 2009, 2011). O problema de pesquisa que se buscou responder foi: Como se caracterizam os conhecimentos sobre Trigonometria e qual a natureza dos conhecimentos relativos a questões sobre a aprendizagem e o ensino de Trigonometria apresentados por professores de matemática que atuam na Educação Básica? A partir disso os objetivos da pesquisa foram: caracterizar os conhecimentos sobre Trigonometria apresentados por professores de matemática que atuam na Educação Básica e; desvelar a natureza desses conhecimentos dos professores de matemática em relação a questões sobre o ensino e a aprendizagem da Trigonometria. Para a realização da pesquisa foi adotada a abordagem qualitativa, de cunho descritivo e explicativo, com os procedimentos metodológicos de análise sustentados pela análise de conteúdo de Bardin (2009). A organização dos dados contou com o auxílio do software Atlas.ti por meio do qual foram elencados o conhecimento de conteúdo da matéria, o conhecimento pedagógico de conteúdo e o conhecimento curricular e as funções de expansão dos discursos propostas por Duval (2004), para posterior análise. Os resultados dessa pesquisa indicam que os conhecimentos docentes que se sobressaíram foram os conhecimentos de conteúdo da matéria a ser ensinada. Esses resultados apontam para a necessidade de um olhar mais atento pelos professores sobre os demais saberes necessários à prática educativa e que parecem não estarem sendo postos em prática.

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Karadeniz, Ilyas. "UCSMP Teachers’ Perspectives when Using Graphing Calculators in Advanced Mathematics." Scholar Commons, 2015. http://scholarcommons.usf.edu/etd/5712.

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Nowadays, technology plays a fundamental role in education, in general, and in mathematics education in particular. The graphing calculator has been an important technological tool in mathematics classrooms since its invention and introduction in 1985 by Casio. As graphing calculators provided so many uses, their contribution to the teaching and learning process has been investigated by many researchers who have shown the use of such technology can have a significant effect on improving mathematics teaching and learning. Investigating the impact of graphing calculators on student learning is important. It is also essential to research teachers’ perspectives on how using graphing calculators in mathematics determines how such use affects their teaching and learning. However, there are few studies on this issue. Therefore, this dissertation study may fill the gap in the literature in terms of examining high school mathematics teachers’ perspectives when they teach a precalculus course with technology integrated in the curriculum materials. In this study, I analyzed eleven teachers’ perspectives about using graphing calculator technology in a precalculus course, titled Functions, Statistics, and Trigonometry (FST). This study was a descriptive intrinsic case study in which I analyzed teachers’ perspectives about how they use graphing calculators in the FST course, specifically about their teaching and students’ learning with available graphing calculator technology. Additionally, I explored teachers' perspectives about the issues they face when using the available technology and for what topics teachers frequently used it. I used mixed methods to examine eleven mathematics teachers’ perspectives about their teaching, students’ learning, and issues that arise when they use graphing calculator technology. In the quantitative part of the study, I created an Index of Teachers’ Initial Perceived Attitude and Experience Level and an Index of Teachers’ Use of Graphing Calculators to measure teachers’ perspectives on technology use at the beginning and end of the school year, respectively. In the qualitative inquiry, I analyzed teachers’ responses to semi-structured interview questions by using thematic analysis. The results of this study showed eight of the eleven mathematics teachers’ students used graphing calculators with Computer Algebra System (CAS) capability loaned by The University of Chicago School Mathematics Project (UCSMP). Five teachers had a high initial perceived attitude and experience level and the other six teachers had a medium level. All teachers reported that helping students learn to use a symbolic manipulator was equally or less important than to use a graphing calculator. The themes (1) Teachers’ use of graphing calculators, (2) Teachers’ opinions about students’ use of graphing calculators, and (3) Teachers’ issues with graphing calculator technology were created to explain teachers’ responses to interview questions related to their graphing calculator perspectives throughout the year. Teachers typically used graphing calculators almost every day for such purposes as exploring mathematics, solving problems, and checking work. Some teachers reported the benefits of using graphing calculators in terms of instruction were focusing on the concepts and showing additional solution approaches. Teachers who wanted their students to be able to do some work without graphing calculators used no calculator tests or questions on which graphing calculators were not allowed as part of their assessment process. Teachers mentioned the need for a manual showing the steps for using graphing calculators with CAS. Teachers’ opinions about students’ use of graphing calculators included that students generally liked them. Teachers reported graphing calculators positively affected students’ learning because students were able to find the answers for problems and have better visualization opportunities. However, teachers reported some meaning was missing and students’ arithmetic skills were negatively affected because of the presence of graphing calculators. Additionally, five teachers indicated their students relied on the graphing calculators too much. The most common issue teachers had relative to graphing calculator technology was the liability issue of the graphing calculators sent by UCSMP for students to loan. Teachers were responsible for those loaned graphing calculators. Additionally, cheating, using features that minimized the mathematics, and not being familiar with the type of graphing calculators loaned from UCSMP were other issues teachers reported. Teachers’ graphing calculator use was demonstrated based on the index of teachers’ use of graphing calculators. Seven teachers were high in terms of their use of graphing calculators at the end of the school year and four teachers had a medium use of graphing calculators. For implications of this study, mathematics teacher educators can use the results to improve professional development programs for teachers. They might create workshops based on teachers’ perspectives and their initial perceived attitude and experience level. Additionally, textbook developers can create more exploration activities with graphing calculators, especially with CAS.

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Depizoli, Carlos Antonio. "Matemática e música e o ensino de funções trigonométricas." Universidade Tecnológica Federal do Paraná, 2015. http://repositorio.utfpr.edu.br/jspui/handle/1/1361.

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CAPES
Utilizando-se da relação entre a Matemática e a Física e tendo a música como fonte de recursos associados a conceitos matemáticos, este trabalho objetiva contribuir com o desenvolvimento de habilidades nos estudantes na aprendizagem de conteúdos matemáticos como as funções trigonométricas. Conceitos importantes relacionados à acústica são apresentados, mostrando a importância da Série de Fourier Contínua no desenvolvimento de sintetizadores analógicos aditivos e seu uso no ensino de trigonometria.
Using the relationship between Mathematics and Physics and having music as a source of features associated with mathematical concepts, this work aims to contribute to the development of skills in students who are learning mathematical subjects such as trigonometric functions. Important concepts related to acoustics are presented, showing the importance of Continuous Fourier Series (CFS) in the development of Additive Synthesizer and its use in trigonometry.

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Ress, David Andress. "Development of Fuzzy Trigonometric Functions to Support Design and Manufacturing." NCSU, 2010. http://www.lib.ncsu.edu/theses/available/etd-03112010-104230/.

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It is a well established fact that design undergoes stages from imprecision to precision. In the early design stages, fuzzy logic is a natural tool for modeling since it is by definition an imprecise representation. The mathematics behind fuzzy numbers have been well developed and defined in literature; yet, very little research exists in the form of fuzzy trigonometric functions. Two design problems are presented to support the motivation behind this research followed by a review of fuzzy set theory. Several approaches for mapping Y = cos(X) into the fuzzy realm are then discussed followed by the development of special purpose fuzzy trigonometric functions and fuzzy inverse trigonometric functions which are computationally simple and easy to implement. With these functions, 8 forward and 6 inverse trigonometric identities are shown to exist in the fuzzy realm. The proposal concludes by examining three engineering problems. The first problem involves the design of a fuzzy truss bridge with fuzzy forces. The second problem analyzes fuzzy forces on a block positioned on an inclined plane. The last example utilizes the fuzzy inverse trigonometric functions to calculate fuzzy bond angles within a chemical compound.

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Javed, Mohsin. "Algorithms for trigonometric polynomial and rational approximation." Thesis, University of Oxford, 2016. https://ora.ox.ac.uk/objects/uuid:23a36d72-0299-4c63-98e8-d0aa088c062e.

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This thesis presents new numerical algorithms for approximating functions by trigonometric polynomials and trigonometric rational functions. We begin by reviewing trigonometric polynomial interpolation and the barycentric formula for trigonometric polynomial interpolation in Chapter 1. Another feature of this chapter is the use of the complex plane, contour integrals and phase portraits for visualising various properties and relationships between periodic functions and their Laurent and trigonometric series. We also derive a periodic analogue of the Hermite integral formula which enables us to analyze interpolation error using contour integrals. We have not been able to find such a formula in the literature. Chapter 2 discusses trigonometric rational interpolation and trigonometric linearized rational least-squares approximations. To our knowledge, this is the first attempt to numerically solve these problems. The contribution of this chapter is presented in the form of a robust algorithm for computing trigonometric rational interpolants of prescribed numerator and denominator degrees at an arbitrary grid of interpolation points. The algorithm can also be used to compute trigonometric linearized rational least-squares and trigonometric polynomial least-squares approximations. Chapter 3 deals with the problem of trigonometric minimax approximation of functions, first in a space of trigonometric polynomials and then in a set of trigonometric rational functions. The contribution of this chapter is presented in the form of an algorithm, which to our knowledge, is the first description of a Remez-like algorithm to numerically compute trigonometric minimax polynomial and rational approximations. Our algorithm also uses trigonometric barycentric interpolation and Chebyshev-eigenvalue based root finding. Chapter 4 discusses the Fourier-Padé (called trigonometric Padé) approximation of a function. We review two existing approaches to the problem, both of which are based on rational approximations of a Laurent series. We present a numerical algorithm with examples and compute various type (m, n) trigonometric Padé approximants.

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SOUZA, Luciano. "New trigonometric classes of probabilistic distributions." Universidade Federal Rural de Pernambuco, 2015. http://www.tede2.ufrpe.br:8080/tede2/handle/tede2/5127.

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In this thesis, four new probabilistic distribution classes are presented and investigated: sine, cosine, tangent and secant. For each of which a new kind of distribution was created, which were used for modelling real life data.By having an exponential distribution to compare the biases, a numerical simulation was obtained, making it possible to verify that the bias tends to zero as the sample size is increased. In addition to that, some numerical results for checking maximum likelihood estimates, as well as the results for finite samples, were obtained, just as much as several class properties and their respective distributions were also obtained, along with the expansions, maximum likelihood estimates, Fisher information, the first four moments, average, variance, skewness, and kurtosis, the generating function of moments and Renyi’s entropy. It was evidenced that all distributions have shown good fit when applied to real life data, when in comparison to other models. In order to compare the models, the Akaike Information Criterion (AIC), the Corrected Akaike Information Criterion (CAIC), the Bayesian Information Criterion (BIC), the Hannan Quinn Information Criterion (HQIC) were used, along with two other main statistic sources: Cramer-Von Mises and Anderson-Darling. As a final step, the results of the analyses and the comparison of the results are brought up, as well as a few directions for future works.
Nesta tese apresentamos e investigamos quatro novas classes trigonométricas de distribuições probabilísticas. As classes seno, cosseno, tangente e secante. Para cada uma das novas classes foi criada uma nova distribuição. Estas quatro novas distribuições foram usadas na modelagem de dados reais. Obtivemos uma simulação numérica, usando como base a distribuição exponencial, para se comparar os vicios (bias) e verificamos que, a medida que aumentamos o tamanho da amostra, o bias tende a zero. Alguns resultados numéricos para ver estimativas de máxima verossimilhança e os resultados para amostras finitas foram obtidos. Várias propriedades das classes e as suas distribuições foram obtidos. Obtemos as expansões, as estimativas de máxima verossimilhança, informações de Fisher, os quatro primeiros momentos, média, variância, assimetria e curtose, a função geradora de momentos e a entropia Rényi. Mostramos que todas as distribuições têm proporcionado bons ajustes quando aplicadas a dados reais, em comparação com outros modelos. Na comparação dos modelos foram utilizados: o Akaike Information Criterion (AIC), o Akaike Information Criterion Corrigido (CAIC), a informação Bayesian Criterion (BIC), o critério de informação Hannan Quinn (HQIC) e duas das principais estatísticas também foram utilizadas: Cramer -von Mises e Anderson-Darling. Por fim, apresentamos os resultados da análise e comparação dos resultados, e orientações para trabalhos futuros.

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Siyepu, Sibawu Witness. "Analysis of errors in derivatives of trigonometric functions: a case study in an extended curriculum programme." Thesis, University of the Western Cape, 2012. http://hdl.handle.net/11394/5207.

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Philosophiae Doctor - PhD
The purpose of this study was to explore errors that are displayed by students when learning derivatives of trigonometric functions in an extended curriculum programme. The first aim was to identify errors that are displayed by students in their solutions through the lens of the APOS theory. The second aim was to address students' errors by using the two principles of Vygotsky's socio-cultural theory of learning, namely the zone of proximal development and more knowledgeable others. The research presented in this thesis is a case study located in the interpretive paradigm of qualitative research. The participants in this study comprised a group of students who registered for mathematics in the ECP at Cape Peninsula University of Technology, Cape Town, South Africa. The study was piloted in 2008 with a group of twenty students who registered for mathematics in the ECP for Chemical Engineering. In 2009 thirty students from the ECP registered for mathematics in Chemical Engineering were selected to participate in the main study. This study was conducted over a period of four and a half years. Data collection was done through students' written tasks; classroom audio and video recordings and indepth interviews. Data were analysed through categorising errors from students' written work, and finding common themes and patterns in audio and video recordings and from the in-depth interviews. The findings of this study revealed that students committed interpretation, arbitrary, procedural, linear extrapolation and conceptual errors. Interpretation errors arise when students fail to interpret the nature of the problem correctly owing to over-generalisation of certain mathematical rules. Arbitrary errors arise when students behave arbitrarily and fail to take account of the constraints laid down in what is given. Procedural errors occur when students fail to carry out manipulations or algorithms although they understand concepts in problem. Linear extrapolation errors happen through an overgeneralisation of the property f (a + b) = f (a) + f (b) , which applies only when f is a linear function Conceptual errors occur owing to failure to grasp the concepts involved in the problem or failure to appreciate the relationships involved in the problem. The findings were consistent with literature indicated that errors are based on students’ prior knowledge, as they over-generalise certain mathematical procedures, algorithms and rules of differentiation in their solutions. The use of learning activities in the form of written tasks; as well as classroom audio and video recordings assisted the lecturer to identify and address errors that were displayed by students when they learned derivatives of trigonometric functions. The students claimed in their interviews that they benefited from class discussions as they obtained immediate feedback from their fellow students and the lecturer. They also claimed that their performances improved as they continued to practice with the assistance of more knowledgeable students, as well as the lecturer. This study supports the view from the literature that identification of errors has immense potential to address students’ poor understanding of derivatives of trigonometric functions. This thesis recommends further research on errors in various sections of Differential Calculus, which is studied in an extended curriculum programme at Universities of Technology in South Africa.

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Supe, Tushar. "Super - cordic: Low delay cordic architectures for computing complex functions." Thesis, Georgia Institute of Technology, 2015. http://hdl.handle.net/1853/54475.

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This thesis proposes an optimized Co-ordinate Rotation Digital Computer (CORDIC) algorithm in the rotation and extended vectoring mode of the circular co-ordinate system. The CORDIC algorithm computes the values of trigonometric functions and their inverses. The proposed algorithm provides the result with a lower overall latency than existing systems. This is done by using redundant representations and approximations of the required direction and angle of each rotation. The algorithm has been designed to provide the result in a fixed number of iterations $n$ for the rotation mode and $3\lceil n/2 \rceil + \lfloor n/2 \rfloor$ for the extended vectoring mode; where, $n$ is a design parameter. In each iteration, the algorithm performs between 0 and $p/n$ parallel rotations, where, $p$ is the number of precision bits and $n$ is the selected number of iterations. A technique to handle the scaling factor compensation for such an algorithm is proposed. The results of the functional verification for different values of $n$ and an estimation of the overall latency are presented. Based on the results, guidelines to choosing a value of $n$ to meet the required performance have also been presented.

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Marchi, Dominic J. "A Study of Student Understanding of the Sine Function through Representations and the Process and Object Perspectives." The Ohio State University, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=osu1343253667.

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Iochucki, Suellen Karina Palhano. "PROPOSTAS PARA O ENSINO DA TRIGONOMETRIA:INTRODUÇÃO À APROXIMAÇÃO DE FUNÇÕES PERIÓDICAS POR POLINÔMIOS TRIGONOMÉTRICOS." UNIVERSIDADE ESTADUAL DE PONTA GROSSA, 2016. http://tede2.uepg.br/jspui/handle/prefix/1512.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
This work presents an approach to trigonometry content using Geogebra and Maxima programs. It provides a road map of how trigonometry can be worked in the classroom divided into stages . It proposes applications of trigonometry in different areas of knowledge and suggests the introduction of the approach of periodic functions by trigonometric polynomials . It is a qualitative and exploratory research . Its relevance is justified because it allows the teacher to look at the process of teaching and learning significantly to the student. During the development of research, the approach of trigonometry content was the history of mathematics and the use of software. The use of computing resources was an important ally to scientific knowledge, as well as the use of applications for the contextualization.
Este trabalho apresenta uma abordagem sobre o conteúdo de Trigonometria utilizando os programas Geogebra e Maxima. Traz um roteiro de como a Trigonometria pode ser trabalhada em sala de aula dividido em etapas. Propõe aplicações da Trigonometria em diferentes áreas do conhecimento e sugere a introdução da aproximação de funções periódicas por polinômios trigonométricos. Trata-se de uma pesquisa qualitativa e exploratória. A sua relevância justificase por possibilitar ao professor olhar o processo de ensino-aprendizagem de forma significativa ao aluno. Durante o desenvolvimento da pesquisa, a abordagem do conteúdo de trigonometria ocorreu pela história da matemática e com a utilização de softwares. A utilização dos recursos computacionais mostrou-se importante aliada ao saber científico, assim como a utilização de aplicações para a contextualização do tema.

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Qi, Hui, University of Western Sydney, of Science Technology and Environment College, and School of Computing and Information Technology. "Multi-polynomial higher order neural network group models for financial data and rainfall data simulation and prediction." THESIS_CSTE_CIT_Qi_H.xml, 2001. http://handle.uws.edu.au:8081/1959.7/343.

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Multi-Polynomial Higher Order Neural Network Group Models (MPHONNG) program developed by the author will be studied in this thesis. The thesis also investigates the use of MPHONNG for financial data and rainfall data simulation and prediction. The MPHONNG is combined with characteristics of Polynomial function, Trigonometric polynomial function and Sigmoid polynomial function. The models are constructed with three layers Multi-Polynomial Higher Order Neural Network and the weights of the models are derived directly from the coefficents of the Polynomial form, Trignometric polynomial form and Sigmoid polynomial form. To the best of the authors knowledge, it is the first attempt to use MPHONNG for financial data and rainfall data simulation and prediction. Results proved satisfactory, and confirmed that MPHONNG is capable of handling high frequency, high order nonlinear and discontinuous data.
Master of Science (Hons)

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21

Bertoni, Vanessa. ""Funções e polinômios univalentes: algumas propriedades e aplicações"." Universidade de São Paulo, 2006. http://www.teses.usp.br/teses/disponiveis/55/55134/tde-11092006-162445/.

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O objetivo principal deste trabalho é o estudo das funções univalentes e de suas propriedades. Este estudo é direcionado principalmente aos polinômios univalentes e à investigação de prolemas extremos envolvendo seus coeficientes, seus zeros e suas propriedades geométricas. Encontramos uma relação interessante entre os polinômios univalentes e os polinômios univalentes definida por Suffridge. Geramos várias funções univalentes estreladas e convexas através de suas propriedades geométricas e da localização de seus zeros.
The main aim of this work is the study of univalent functions and their properties. This study is focused speciall on the univalent polynomials and theinvestigation of extremal problems involving their coefficients, zeros and geometric properties. We find a very ineresting relation between univalent polynomials through a class of univalent starlike and convex functions involving their geometric properties and the location of their zeros.

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Maia, Joaildo. "O ensino de fun??es trigonom?tricas atrav?s do software geogebra." Universidade Federal do Rio Grande do Norte, 2013. http://repositorio.ufrn.br:8080/jspui/handle/123456789/18659.

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Universidade Federal do Rio Grande do Norte
In this study, we sought to address the weaknesses faced by most students when they were studying trigonometric functions sine and cosine. For this, we proposed the use of software Geogebra in performing a sequence of activities about the content covered. The research was a qualitative approach based on observations of the activities performed by the students of 2nd year of high school IFRN - Campus Caicfio. The activities enabled check some diculties encountered by students, well as the interaction between them during the tasks. The results were satisfactory, since they indicate that the use of software contributed to a better understanding of these mathematical concepts studied
Nesse trabalho, buscou-se suprir as defi ci?ncias enfrentadas por boa parte dos alunos quando eram estudadas as funfi??es trigonomfietricas seno e co-seno. Para isso, prop?s-se a utilizafi??o do software Geogebra na realizafi??o de uma sequ?ncia de atividades sobre o contefiudo abordado. A pesquisa teve uma abordagem qualitativa, baseada em observa fi??es das atividades realizadas pelos alunos do 2o Ano do Ensino Mfi?dio do IFRN { Campus Caicfi?. As atividades possibilitaram verificar algumas dificuldades encontradas pelos alunos, assim como a interafi??o existente entre eles durante a realizafi??o das tarefas. Os resultados obtidos foram satisfatfi?rios, pois indicam que a utilizafi??o do software contribuiu para uma melhor compreens?o desses conceitos matemfi?ticos estudados

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23

Dias, Maria de Fátima Castilho. "Uma abordagem para análise, classificação e resolução de problemas que envolvem trigonometria : exemplos de aplicação." Master's thesis, Universidade de Évora, 2010. http://hdl.handle.net/10174/20921.

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Como professora de matemática, com experiência no Ensino Secundário, tenho constatado que os alunos revelam bastantes dificuldades de compreensão, aquisição e aplicação dos conteúdos trigonométricos. Também os conteúdos de trigonometria dos actuais manuais escolares do ensino secundário não são suficientes para aqueles alunos que pretendem seguir estudos superiores nas áreas da Física, da Matemática, da Engenharia, Este trabalho destina-se essencialmente a esse grupo de alunos, mas também pode servir de material de apoio a professores do Ensino Básico/Secundário que queiram enriquecer e aprofundar os seus conhecimentos nesta área. Ao longo deste trabalho, propomos uma forma alternativa de apresentação deste tema, pensamos que deste modo os alunos possam compreender "como, donde e porquê" aparecem as relações trigonométricas. Classificamos e analisamos em profundidade equações trigonométricas, com a finalidade de propor abordagens diferentes de resolução, umas típicas outras originais. Analisamos desigualdades trigonométricas e sistemas de equações trigonométricas utilizando métodos de resolução distintos. Apresentamos versões distintas de demonstrações para as mesmas afirmações, e apresentamos problemas de Geometria, de Topografia e de Física, mostrando desta forma a aplicação prática da Trigonometria. Finalizamos este trabalho com a apresentação de algumas relações trigonométricas na teoria dos Números Complexos. – ABSTRACT: School students, I have noticed that students have many difficulties in understanding trigonometric contents. The contents related to trigonometry that are presented, in modem textbooks adopted by Secondary Schools are not enough for those students who intend to take a degree in areas in field of physics, mathematics and engineering. This study is essentially intended for the above-mentioned students, but it can also be used as a support material for a secondary school teachers who want to enrich and increase their scientific knowledge in the field of trigonometry. This project suggests an alternative way to introduce the topic so that student can understand "how, from, where and why" trigonometric relationships appear. Trigonometric equations are classified and examined deeply in order to suggest different approaches for their solution, typical ones and original others. Trigonometric inequalities and systems of trigonometric equations of special types are also analysed using different solution methods. Different versions of demonstrations for the same statements are presented as well as problems of Geometry, Topography and Physics, with the purpose of showing the practical use of trigonometry. This project ends with the presentation of some trigonometric relationships in the theory of Complex Numbers.

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Lemos, Paulo Giovane Aparecido. "Funções aplicadas a física e química." Universidade Federal de Juiz de Fora (UFJF), 2013. https://repositorio.ufjf.br/jspui/handle/ufjf/3433.

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CAPES - Coordenação de Aperfeiçoamento de Pessoal de Nível Superior
Neste trabalho apresentamos uma sequência de atividades utilizando os conceitos de alguns tipos funções: afim, logarítmica e trigonométricas. Tratando, também, a interdisciplinaridade com as disciplinas física e química. Nestas atividades serão construídas tabelas com informações sobre duas ou mais grandezas e, posteriormente, representações gráficas com o auxilio do Excel ou do Geogebra. A atividade sobre a função afim deve ser aplicada aos alunos do 9º ano do ensino fundamental ou alunos do 1º ou 3º ano do ensino médio, esta atividade visa ao aprendizado dos alunos, usando os conceitos de movimentos da física e, assim, mostrando aplicação das funções. Nesta atividade, o aluno deve construir um dispositivo pratico para coletar dados sobre posição e tempo do movimento de um móvel e este deve se aproximar do movimento retilíneo uniforme de um móvel. Com este dispositivo vamos fazer uma filmagem do movimento de um móvel, assim teremos maior facilidade para coletarmos as posições de acordo com tempo, e construir uma tabela. Com a tabela vamos usar o Excel e o Geogebra para construir o gráfico. Com a intervenção do professor de física, devemos chegar ao estudo de uma função afim ao estudo de uma reta em geometria analítica. A atividade logarítmica é sobre a aplicação do logaritmo no cálculo do pH de uma solução. Nesta atividade é acrescentado gradativamente base (HCl) a um ácido (NaOH). A verificação do pH da solução é feito com a fita de titulação e a constatação é feita a partir da função do pH que é pH = -log[H+] ou pOH = -log [OH-] → pH = 14 – pOH, com estas informações é construído uma tabela com informações sobre o volume de ácido, volume da base, o volume da solução e o pH. Com esta tabela construímos o gráfico do pH em função do volume de base usando o Excel ou Geogebra. Com esta atividade podemos também trabalhar noções intuitivas de limite quando o pH está próximo de 7 utilizando as duas fontes, a tabela e o gráfico e descobrindo até mesmo funções de correção da equação do pH. Esta atividade pode ser trabalhada com alunos do 1º ou 2º ano do ensino médio com a intervenção do professor de química quanto aos conceitos químicos aplicados nesta atividade. A atividade sobre funções trigonométricas tenta mostrar que é a função trigonométrica é melhor função para um estudo de movimentos periódicos ou qualquer estudo que envolva periodicidade. Nesta atividade vamos usar o software Tracker para coletarmos informações sobre as posições de um pêndulo simples em relação a sua projeção na horizontal e vertical de acordo com o tempo. O software Tracker é de grande ajuda nesta atividade para filmagem de múltiplas posições que é o que ocorre neste experimento. Esta atividade vem ao encontro do que propõe o PCNEM, pois se refere à função trigonométrica com a função periódica e não à parte algébrica que as identidades trigonométricas aborda. Todas as atividades estão de acordo com os propósitos do PCNEM, agem do CBC/Matemática - SEE/MG e trabalham a interdisciplinaridade entre Matemática e Física ou entre Matemática e Química, mostrando que a Matemática não se trata de uma ciência isolada como tantos alunos pensam.
On this workshop, we will show a few activities using ideas from linear function, logarithm function and trigonometric function ( this one will be associated with chemistry and physics themes) . Activities abording linear function must be applied to students from 9th year from the basic education or 1st and 3rd year from the high school students. First linear function wants to focus the main ideas of movements on physics and showing its applications on functions. On this activity , the student should build na easy way to collect informations about position and time from a movement of a mobile and this one must be the nearest possible from the uniform rectilinear motion. With this device, we are going to make a film of the movement of a mobile, so then we can build a table with all the information we need. With the table we can build graphs using programs such as Excel and Geogebra. Assisted by the physics teacher, we are supposed to make some conclusions about the study of the linear function and the straight on analytic geometry Logarithm function is used to calculate the pH of a chemistry solution. The solution will change its pH if basis(NaOH) or acid(HCl) be increased to it according to this function pH = -log[H+] ou pOH = -log [OH-] → pH = 14 – pOH, with this informations we can build a pH table in function of the volume and then a graph can be constructed using Excel or Geogebra. This activity can also work intuitive notions of limit when the pH close to 7 this through both the table and the graph and finding even functions of pH correction equation. This activity should be worked with students in the 1st or 2nd year of high school with teacher intervention chemistry as applied to chemical concepts in this activity. The trigonometric functions trys to show that it is the best function to a study of periodic movements or any study that involves periodicity. For this function we will use Tracker software to collect informations about the positions of a simple pendulum in relation to its projection in horizontal and vertical according to the time. Tracker is very helpful because it can film a lot of positions that occurs on this experience. All work activities interdisciplinarity between mathematics and physics or between mathematics and chemistry and so showing that mathematics is not just an isolated sciences as many students think.

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Tutkienė, Simona. "Puasono dvimatės lygties vidinių reikšmių uždavinio sprendimas „tilto“ funkcijų metodais." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2011. http://vddb.laba.lt/obj/LT-eLABa-0001:E.02~2011~D_20110803_092027-00523.

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Magistro darbe matematiniu modeliavimu nagrinėjamas Puasono lygties sprendimo efektyvumas naujais metodais. Šiame darbe siūloma spręsti šias lygtis naudojant vadinamąsias „tilto“ funkcijas. Bandomos dviejų rūšių „tilto“ funkcijos: hiperbolinio tangento ir trigonometrinės. Puasono lygties sprendinys ieškomas per „tilto“ funkcijų ir polinomų sandaugų sumą.
In this study Poisson function is solved using “bridge” functions method, meaning that all range is divided to separate zones (“bridges”) and to separate approximation polynomial multiplied of “bridge” functions. Common solution is equal to the sum of separate polynomial multiplied of “bridge” functions. To solve Poisson equation, the so-called "bridge" function was used. Differential equation, the solution we were looking via the "bridge" functions and products of powers of polynomials amount.

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Santos, Jonas José Cruz dos. "Estudo e aplicações das funções hiperbólicas." Universidade Federal da Paraíba, 2015. http://tede.biblioteca.ufpb.br:8080/handle/tede/9335.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
This work intends to show the hyperbolic functions, analyzing their similarities and contrasts with the circular trigonometric functions. To this, we start showing a short review about the circular trigonometry and hyperbole, describing their main elements and properties. Then, we made a study about hyperbolic functions, with the de nitions of sine, cosine and the other hyperbolic functions and their main properties. We nished this work with some applications of these functions on everyday.
Este trabalho tem como objetivo apresentar as funções hiperbólicas, analisando suas semelhanças e diferenças com as funções trigonométricas circulares. Para tanto, iniciamos apresentando uma breve revisão sobre a trigonometria circular e a hipérbole, descrevendo seus principais elementos e propriedades. Posteriormente, realizamos um estudo sobre as funções hiperbólicas, apresentando as de nições do seno, cosseno e das demais funções hiperbólicas e suas principais propriedades. Concluí- mos com algumas aplicações destas funções no cotidiano.

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Tavares, Wellington Silva. "O ensino das funções trigonométricas com o auxílio do software matemático de ambiente grá fico WINPLOT." Universidade Federal de Goiás, 2013. http://repositorio.bc.ufg.br/tede/handle/tede/3791.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
One of the main causes of the great di culty of learning mathematics is the lack of the interest from the students. Aiming to seek an improvement in teaching in order to make students more motivated, this project was developed, which proposes a di erent methodology in the teaching of trigonometric functions with the utilization of mathematic software Winplot. Activities were developed for application of Winplot which might provide to the teacher a more e cient method in the teaching of trigonometric functions. The use of computers in mathematics education aims to make young people be motivated to learn this discipline, changing the routine of class and arousing the interest of the student involved. The Mathematics has undergone signi cant changes in their teaching methods, thus develop logical reasoning, encourage independent thinking, creativity and problem-solving skills became essential to all citizen who wishes to have an active and critical participation in society. Therefore, the utilization of informatics in teaching mathematics o ers these characteristics in a simple and fun without leaving to encourage the student in their learning.
Uma das principais causas da grande di culdade de aprender Matemática é a falta de interesse dos alunos. Visando buscar uma melhoraria para o ensino, a m de tornar os alunos mais motivados, foi desenvolvido este projeto, que propõe uma metodologia diferenciada no ensino das funções trigonométricas com a utilização do software matemático Winplot. Foram desenvolvidas atividades para aplicação do Winplot que poderão proporcionar ao professor um método mais e ciente no ensino das funções trigonom étricas. O uso da informática no ensino da matemática tem o objetivo de fazer com que os jovens se motivem a aprender esta disciplina, mudando a rotina da classe e despertando o interesse do aluno envolvido. A matemática vem sofrendo mudanças signi cativas em seus métodos de ensino, assim desenvolver o raciocínio lógico, estimular o pensamento independente, a criatividade e a capacidade de resolver problemas tornou-se essencial a todo cidadão que deseja ter uma participação ativa e crítica na sociedade. Logo, a utilização da informática no ensino da matemática oferece essas características de uma maneira simples e divertida sem deixar de estimular o aluno em sua aprendizagem.

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Souza, Jakson Idernando Gonzaga de. "UtilizaÃÃo do software GeoGebra no ensino das funÃÃes trigonomÃtricas." Universidade Federal do CearÃ, 2014. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=12595.

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nÃo hÃ
Discutir o uso pedagÃgico de recursos tecnolÃgicos nas aulas de MatemÃtica tem sido foco de muitos estudos, uma vez que por si sÃ eles nÃo garantem um novo modelo educacional. Neste sentido, esse trabalho tem o objetivo de explorar o software GeoGebra como recurso pedagÃgico no ensino das funÃÃes trigonomÃtricas para alunos da 2Â sÃrie do Ensino MÃdio, de uma escola da rede particular no estado do CearÃ, atravÃs da visualizaÃÃo das funÃÃes trigonomÃtricas no cÃrculo trigonomÃtrico, suas representaÃÃes grÃficas e o papel dos parÃmetros na construÃÃo dos grÃficos dessas funÃÃes, oportunizando o contato com recursos diferentes dos presentes em sala de aula. Esse trabalho mostra que o uso da simulaÃÃo computacional pode ser um meio eficiente de promover o aprendizado de forma significativa, facilitando assim o desenvolvimento de habilidades na compreensÃo por parte dos discentes dos conteÃdos relacionados Ãs funÃÃes trigonomÃtricas, num ambiente onde os conteÃdos abordados sÃo trabalhados utilizando o amplo campo de informaÃÃes e interaÃÃes do computador.
Discuss the pedagogical use of technological resources in mathematics classes has been the focus of many studies, since they alone do not guarantee a new educational model. In this sense, this work aims to explore the GeoGebra software as a teaching resource in the teaching of the trigonometric functions for students of 2nd year of high school, a school of private schools in the state of CearÃ, through visualization of the trigonometric functions in the unit circle , their graphical representations and the role of parameters in the construction of the graphs of these functions, providing the opportunity for contact with the different features present in the classroom. This work shows that the use of computer simulation can be an effective means of promoting learning significantly, thus facilitating the development of skills in understanding by students of the content related to trigonometric functions, in an environment where the subjects covered are worked using the broad field of information and computer interactions.

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Li, Kai, Heinz Rüdiger, Rocco Haase, and Tjalf Ziemssen. "An Innovative Technique to Assess Spontaneous Baroreflex Sensitivity with Short Data Segments: Multiple Trigonometric Regressive Spectral Analysis." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2018. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-233899.

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Objective: As the multiple trigonometric regressive spectral (MTRS) analysis is extraordinary in its ability to analyze short local data segments down to 12 s, we wanted to evaluate the impact of the data segment settings by applying the technique of MTRS analysis for baroreflex sensitivity (BRS) estimation using a standardized data pool. Methods: Spectral and baroreflex analyses were performed on the EuroBaVar dataset (42 recordings, including lying and standing positions). For this analysis, the technique of MTRS was used. We used different global and local data segment lengths, and chose the global data segments from different positions. Three global data segments of 1 and 2 min and three local data segments of 12, 20, and 30 s were used in MTRS analysis for BRS. Results: All the BRS-values calculated on the three global data segments were highly correlated, both in the supine and standing positions; the different global data segments provided similar BRS estimations. When using different local data segments, all the BRS-values were also highly correlated. However, in the supine position, using short local data segments of 12 s overestimated BRS compared with those using 20 and 30 s. In the standing position, the BRS estimations using different local data segments were comparable. There was no proportional bias for the comparisons between different BRS estimations. Conclusion: We demonstrate that BRS estimation by the MTRS technique is stable when using different global data segments, and MTRS is extraordinary in its ability to evaluate BRS in even short local data segments (20 and 30 s). Because of the non-stationary character of most biosignals, the MTRS technique would be preferable for BRS analysis especially in conditions when only short stationary data segments are available or when dynamic changes of BRS should be monitored.

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Curi, Neto Emilio. "Aplicação do polinômio de Taylor na aproximação da função Seno." Universidade Federal de Goiás, 2014. http://repositorio.bc.ufg.br/tede/handle/tede/3493.

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Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES
In this work the main goal is focused on applying the theory of Taylor polynomial approximations applied on the trigonometric function defined by f : [0; 2 ] 􀀀! R, where f(x) = sin(x). To achieve this goal, eight sections were developed, in which initially a reflection on the problem and the need to obtain the values in this respect in that it is wide angle measure x is presented. Is presented and subsequently treated a problem involving the movement of a pendulum, which uses the approximation sin(x) x where x belongs to a certain range. In the sections that follow a literature review of the theories of differential and integral calculus is presented, and the related theory of Taylor approximation of functions by polynomials. Later we used these theories to analyze and determine polynomials approximating the function f(x) = sin(x) in a neighborhood of the point x = 0, and estimate the error when we applied these approaches. At this time the error occurred due to the approach used in the pendulum problem was also analyzed. Finally a hint of practice to be held in the classroom using the theories treated here as well as the study of the problem of heat transfer in a bar through the theory of Fourier activity is presented.
Neste trabalho o objetivo principal está focado em aplicar a teoria de Taylor relativa à aproximações polinomiais aplicadas à função trigonométrica definida por f : [0; 2 ] 􀀀! R, onde f(x) = sen(x). Para alcançar esse objetivo, foram desenvolvidas oito seções, nas quais inicialmente é apresentada uma reflexão sobre a necessidade e a problemática de obtêr-se os valores desta relação a medida em que varia-se a medida do ângulo x. Posteriormente é apresentado e tratado um problema envolvendo o movimento de um pêndulo, o qual utiliza a aproximação sen(x) x onde x pertence o um certo intervalo. Nas seções que seguem é apresentada uma revisão bibliográfica das Teorias do Cálculo Diferencial e Integral, assim como da Teoria de Taylor relacionada à aproximação de funções através de polinômios. Posteriormente utilizou-se estas teorias para analisar e determinar polinômios que aproximam a função sen(x) em uma vizinhança do ponto x = 0, assim como estimar o erro gerado ao utilizar-se estas aproximações. Nesse momento também foi analisado o erro ocorrido devido à aproximação utilizada no problema do pêndulo. Por fim é apresentada uma sugestão de atividade prática a ser realizada em sala de aula utilizando as teorias aqui tratadas, assim como o estudo do problema de transferência de calor em uma barra através da teoria de Fourier.

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Abrantes, Wagner Gomes Barroso. "A função periódica para o ensino médio." Universidade Federal do Amazonas, 2015. http://tede.ufam.edu.br/handle/tede/4254.

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The periodic functions are a topic of enormous importance for the High School. This Labor Completion of course covers the basics for understanding the topic, the characteristics of these functions and those with whom students have contact in basic education. The use of computational resources in the teaching of Periodic Functions and their applications are also part of the research.
As funções periódicas constituem um tema de enorme relevância para o Ensino Médio. Este Trabalho de Conclusão de Curso aborda os conceitos básicos para a compreensão do tema, as características dessas funções e aquelas com as quais os alunos tem contato na educação básica. O emprego dos recursos computacionais no ensino das Funções Periódicas e suas aplicações também fazem parte da pesquisa.

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Costa, Tiago Bezerra da. "Funções trigonométricas com o auxílio do GeoGebra /." São José do Rio Preto, 2019. http://hdl.handle.net/11449/181763.

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Orientador: Rita de Cássia Pavan Lamas
Banca: Flavia Souza Machado da Silva
Banca: Durval José Tonon
Resumo: Uma das aplicações das funções trigonométricas é a modelagem de alguns fenômenos periódicos, são exemplos de tais fenômenos, o movimento da luz solar, as fases da Lua, a circulação do sangue e o movimento dos planetas. Abordamos aqui aspectos teóricos das funções seno e cosseno, essas são as principais funções trigonométricas trabalhadas no ensino médio. Utilizando um dos materiais oficiais das escolas públicas, o Caderno do Aluno, propusemos atividades sobre as funções trigonométricas a uma turma da 2ª série do Ensino Médio da Rede do Estado de São Paulo, posteriormente analisamos os resultados da nossa aplicação, em sala de aula, de tais atividades. Apresentaremos ainda uma proposta de aula com a utilização do software GeoGebra para a aplicação na sala de informática no intuito de tornar a aprendizagem das funções trigonométricas um pouco mais dinâmica. Analisamos também os resultados dessa aplicação
Abstract: One of the applications of the trigonometric functions is the modeling some periodic phenomenas, examples of such phenomenas are the movement of sunlight, the phases of the Moon, the circulation of blood and the movement of the planets. We address here theoretical aspects of the sine and cosine functions, these are the main trigonometric functions worked out in high school. Using one of the official public school materials, Student Notebook, we proposed activities on the trigonometric functions of high school class from the State of São Paulo. Later, we analyzed the results of our application in the classroom, of such activities. We will present a lesson proposal with the use of GeoGebra software for the application in the computer laboratory in order to make the learning of the trigonometric functions a little more dynamic. We also analyze the results of this application
Mestre

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Ribeiro, Márcia Regina Ramos Costa. "Possibilidades e dificuldades no desenvolvimento de situações de aprendizagem envolvendo funções trigonométricas." Pontifícia Universidade Católica de São Paulo, 2011. https://tede2.pucsp.br/handle/handle/10885.

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Fundo de Apoio à Pesquisa do Estado de São Paulo
This paper is part of an ongoing research on Mathematics in the curriculum, based on the assumption that the characterization of the prior knowledge on Math of the students may help in significant learning. In this perspective, the research focused on the prior knowledge of students in Brazilian Ensino Médio (roughly equivalent to American high school) on trigonometric functions, using qualitative research and participant observation. The objective is to understand the possibilities and difficulties in the use of the material given to students of the schools of the State of São Paulo, focusing in these students prior knowledge on trigonometric functions, identifying obstacles that may rise during these activities and checking the need for interventions to promote the build-up of knowledge on the subject. The proposal was to have students do activities proposed on the material supplied by the state of São Paulo. The analysis is based on Ausubel s theory of significant learning, on Coll s constructivist perspective (2006) and Pozo s studies, concerning learning and teaching contents. The results indicate that the prior knowledge of the students may be divided into many groups, given its important characteristic of including conceptual knowledge as well as procedures, values, regulations and attitudes. The ability to identify this prior knowledge is an important tool to create better pedagogical interventions and significant learning
Esse trabalho está inserido na linha de pesquisa A Matemática na Estrutura Curricular e Formação de Professores e no projeto de pesquisa A aprendizagem significativa e conhecimentos prévios: investigando o currículo de Matemática, em uma perspectiva construtivista; tem como pressuposto que a caracterização dos conhecimentos prévios dos estudantes relacionados aos conteúdos matemáticos que se pretende ensinar contribui para a aprendizagem significativa. Nessa perspectiva, o objetivo desta pesquisa é compreender as possibilidades e dificuldades em utilizar o material distribuído aos alunos da rede pública do Estado de São Paulo, focando conhecimentos prévios desses estudantes em relação ao conteúdo funções trigonométricas, identificando dificuldades que podem surgir durante a execução dessas atividades e verificando as necessidades de intervenções para a promoção da construção de conhecimento relativo ao tema. Para essa investigação, foi utilizada a pesquisa qualitativa e a técnica da observação participante. Foram observadas as ações de um grupo de alunos do segundo ano do Ensino Médio de uma escola pública estadual, durante a realização de atividades propostas e contidas no material. A análise está apoiada na teoria da aprendizagem significativa de Ausubel, na perspectiva construtivista de Coll (2006) e nos estudos de Pozo (2002) relativos aos conteúdos de ensino e à aprendizagem. Os resultados indicam que os conhecimentos prévios dos alunos relacionados as funções trigonométricas podem ser classificados em uma grande variedade de grupos, dada a sua importante característica de incluir tanto conhecimentos conceituais como procedimentos, valores, normas e atitudes; a caracterização desses conhecimentos prévios dos alunos, por parte do professor, a cada nova experiência educativa, constitui-se em importante ferramenta para a realização de intervenções pedagógicas mais eficientes e geradoras de aprendizagem significativa

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Hristov, Sobrinho Dimitrie. "O ensino de funções trigonométricas através da resolução de problemas." Universidade Federal de São Carlos, 2015. https://repositorio.ufscar.br/handle/ufscar/7075.

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Não recebi financiamento
The Brazilian high school, particularly in São Paulo State, has as general objectives the deepening the concepts studied in elementary school, making the student able to continue their studies in higher education and / or vocational courses. In addition, objective the development of critical student's ability to use the mathematical knowledge / concepts studied to understand and solve real situations of their daily lives. However, it is common to see in our schools, students subjected only to the resolution of repetitive exercises, called by George Polya of "routine problems". Consequently, the development of their criticality and their logical / deductive reasoning are affected, not to say, forgotten as goals to be achieved. The research, at the level of Professional Master Degree, reported here was aimed at the "investigation of possible contributions to a methodology based on Problem Solving Theory may have towards the teaching and learning process of trigonometric functions, as well as with the development of mathematical reasoning students of the 2nd year of high school". To this end, it was formulated three issues to be investigated during the course of the research. Are they: (1st) What should be done to generalize the concept of trigonometric ratio (sine, cosine and tangent), studied in the right triangle to the trigonometric cycle? (2nd) How should the transition be held generalization preceded the trigonometric cycle for the study of trigonometric functions in the Cartesian plane in a logical and deductive way? (3rd) What is the contribution that this research brought to my teacher training in math, methodological and didactic perspectives? The investigation results show that the involvement of students in the search for solutions to the problems posed, increased understanding of mathematical concepts worked as well as their application in their everyday situations. Finally, from the my professional training perspective, the research has shown that, when driving the Teaching Process and Learning of Mathematics, from problems "non-routine" and contextualized, the chances of improvement in the understanding and application of mathematical concepts worked as well, interest and involvement of students increased significantly.
O ensino médio brasileiro, em particular no Estado de São Paulo, tem como principais objetivos gerais o aprofundamento dos conceitos estudados no Ensino Fundamental, tornando o aluno capaz de prosseguir com seus estudos no Ensino Superior e/ou em Cursos Profissionalizantes. Além disso, objetiva o desenvolvimento da capacidade crítica do aluno para que esse se utilize dos conhecimentos/conceitos matemáticos estudados para compreender e resolver situações reais de seu cotidiano. Entretanto, é comum observarmos, em nossas escolas, os alunos serem submetidos apenas à resolução de exercícios repetitivos, denominados por George Polya de "problemas rotineiros". Em consequência, o desenvolvimento de sua criticidade e de seu raciocínio lógico/dedutivo ficam prejudicados, para não dizer, esquecidos como objetivos a serem atingidos. A pesquisa, em nível de Mestrado Profissionalizante, aqui relatada tem como objetivo central a “investigação das possíveis contribuições que uma metodologia baseada na Técnica de Resolução de Problemas pode ter para com o processo de ensino e aprendizagem de funções trigonométricas, bem como para com o desenvolvimento do raciocínio matemático de alunos da 2ª série do Ensino médio”. Para tal, foram formuladas três questões a serem investigadas durante o transcorrer da pesquisa. São elas: (1ª) Como deve ser realizada a generalização do conceito de razão trigonométrica (seno, cosseno e tangente), estudado no triangulo retângulo, para o ciclo trigonométrico? (2ª) Como deve ser realizada a transição da generalização procedida do ciclo trigonométrico para o estudo das funções trigonométricas no plano cartesiano, de forma lógica e dedutiva? (3ª) Qual a contribuição que essa pesquisa trouxe para com a minha formação docente, nas perspectivas matemática, didática e metodológica?Os resultados da pesquisa mostram que o envolvimento dos alunos na busca das soluções dos problemas propostos, aumentou a compreensão dos conceitos matemáticos trabalhados, bem como a aplicação deles em situações do cotidiano desses alunos. Finalmente, para minha formação profissional, a pesquisa mostrou que, ao conduzir o Processo de Ensino e Aprendizagem da Matemática, a partir de problemas "não rotineiros" e contextualizados, as chances de melhora na compreensão e aplicação dos conceitos matemáticos trabalhados, bem como, do interesse e do envolvimento dos alunos aumentaram significativamente.

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Ferreira, Edhana das Gra?as. "Uma proposta metodol?gica para o ensino de gr?ficos de fun??es trigonom?tricas." Universidade Federal Rural do Rio de Janeiro, 2017. https://tede.ufrrj.br/jspui/handle/jspui/2408.

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This work aims to propose a methodology that helps teachers on the teaching of graphs of trigonometric functions from the development of studies directed with activities involving the use of technology, in this case the software Graphmatica. It is made a bibliographical review on the historical evolution of Trigonometry, the use of technology and the studies directed in education. Then, are analyzed the chapters on trigonometric functions in the textbooks proposed in the most recent guide, the PNLD 2015. The use of technology is proposed as an alternative to the traditional method of teaching and aims to increase the interest of students as it is something very present in their daily life increasingly dominated by computers, tablets and smartphones. From the graphical representations of the basic functions sine, cosine and tangent, seek the analysis of particularities of the families of functions, such as domain, image and period, through directed activities until the generalization of these specificities with the parameters changes, as well as the graphical transformations that occurred. It is held an exposition of the qualitative results of the application of the material in research conducted in a first year class in high school, which in general aspects were considered positive
Este trabalho tem como objetivo propor uma metodologia que auxilie os professores no ensino de gr?ficos de fun??es trigonom?tricas a partir do desenvolvimento de estudos dirigidos com atividades envolvendo o uso de tecnologia, neste caso o software Graphmatica. ? realizada uma revis?o bibliogr?fica sobre a evolu??o hist?rica da Trigonometria, o uso de tecnologia e dos estudos dirigidos na educa??o. Em seguida, s?o analisados os cap?tulos referentes ?s fun??es trigonom?tricas nos livros did?ticos propostos no guia mais recente, o PNLD 2015. A utiliza??o da tecnologia ? proposta como uma alternativa ao m?todo tradicional de ensino e visa aumentar o interesse dos alunos por se tratar de algo muito presente em seu cotidiano cada vez mais dominado por computadores, tablets e smartphones. A partir das representa??es gr?ficas das fun??es b?sicas seno, cosseno e tangente, busca-se a an?lise de particularidades das fam?lias de fun??es, como dom?nio, imagem e per?odo, atrav?s de atividades direcionadas at? a generaliza??o dessas especificidades com as mudan?as de par?metros, bem como das transforma??es gr?ficas ocorridas. ? realizada uma exposi??o dos resultados qualitativos da aplica??o do material em pesquisa realizada em uma turma de primeira s?rie do ensino m?dio, que em aspectos gerais foram considerados positivos.

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Batista, Valéria Nogueira. "Uma proposta metodológica para o ensino das funções trigonométricas." Universidade Federal de São Carlos, 2015. https://repositorio.ufscar.br/handle/ufscar/7376.

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This research has focused primarily on teaching trigonometric sine and cosine functions. The proposal is to organize the teaching of these functions, in order to improve good learning situations to students, to encourage the transition of trigonometric ratios in right triangle to the trigonometric circle with the use of manipulative materials and GeoGebra software. Based on the theoretical assumptions described in the High School National Curriculum Standards and Curriculum of the State of São Paulo seeking to outline for the teacher, the skills and expertise that student must acquire to be made in basic education, are presented educational alternatives. It aims to help students understand the need and the practical applicability of Trigonometric Functions. Analyzing how these functions are covered in the curriculum documents and books of the National Textbook Program (PNLD 2015), we tried to develop a pioneering work and research including mathematical modeling and taking as reference the exploratory and investigative activities.
Esta pesquisa apresenta como foco principal o ensino das funções trigonométricas seno e cosseno. A proposta é organizar o ensino dessas funções, de forma a potencializar boas situações de aprendizagens aos alunos, que favoreçam a transição das razões trigonométricas no triângulo retângulo para a circunferência trigonométrica com o uso de materiais manipulativos e do software GeoGebra. Partindo dos pressupostos teóricos descritos nos Parâmetros Curriculares Nacionais do Ensino Médio e no Currículo do Estado de São Paulo que procuram delinear para o professor, as habilidades e competências que seu aluno deve adquirir ao ser formado na educação básica, apresentam-se alternativas de ensino que visam contribuir para que os alunos percebam a necessidade e a aplicabilidade prática das Funções Trigonométricas. Analisando como essas funções são abordadas nos documentos curriculares e nos livros do Programa Nacional do Livro Didático (PNLD 2015), procurou-se desenvolver um trabalho de exploração e investigação incluindo a modelagem matemática e tendo como referencial as atividades exploratório-investigativas.

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Oliveira, Luiz Fernando Mosolino de. "Funções trigonométricas." reponame:Repositório Institucional da UFABC, 2016.

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Orientador: Prof. Dr. Daniel Miranda Machado
Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Mestrado Profissional em Matemática em Rede Nacional, 2016.
Este trabalho foi desenvolvido para auxiliar alunos ingressantes no ensino superior, revisando t opicos da trigonometria e de funcoes trigonometricas, podendo auxiliar tambem alunos do ensino medio, professores ou interessados no assunto. Este trabalho parte de estudos iniciais da trigonometria e aborda de maneira simples a construcao de graficos de funcoes trigonometricas da forma f(x) = a+b. sen(c.x+d), onde a, b, c e d sao coeficientes reais que alteram a amplitude, a imagem e período das funcões trigonometricas. Sao deduzidas as formulas de adicao de arcos, que auxiliam na demonstracao de outras equacoes, como por exemplo o teorema das relacoes entre as cordas de circunferencia, de Ptolomeu,. Tambem apresentamos aplicacoes da trigonometria aos triangulos nao retangulos, como a lei dos senos e a lei dos cossenos, utilizada tambem para um triangulo qualquer, auxiliando na demonstracao de equacoes importantes, como por exemplo da força resultante, em Física.
In this work we analyze a simplified version of the Monopoly game using a Markov chain model with discrete time parameter. In the first chapter we discuss on the Classical Theory of Probability, bringing the most important results for this study, preceded by a brief introduction about the ideas of chance throughout the history of mankind and leading thinkers involved in the development of this theory. In the second chapter we make a historical introduction to stochastic processes and Markov chains; then we explain the fundamental concepts of Markov Chains, putting some examples and finally discussing the ergodicity of a Markov chain. In the third chapter, after a brief explanation of the emergence and subsequent evolution of the Monopoly game throughout the twentieth century, we analyze the dynamics of the game by the model of a Markov chain, using as an object of study a simpler version of the game in question.

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Salazar, Denise Mansoldo. "GeoGebra e o estudo das funções trigonométricas no Ensino Médio." Universidade Federal de Juiz de Fora (UFJF), 2015. https://repositorio.ufjf.br/jspui/handle/ufjf/3241.

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Apresentamos uma pesquisa que investiga as potencialidades do software GeoGebra como instrumento tecnológico favorável à aprendizagem das funções trigonométricas no Ensino Médio, a partir das representações gráficas destas funções. Trata-se uma pesquisa caracterizada por uma abordagem qualitativa em que foi usada uma sequência didática de atividades, com tarefas em um nível de exigência crescente. Cada etapa de execução desta pesquisa está respaldada pelos pressupostos teóricos da Engenharia Didática, metodologia adotada em pesquisas que envolvem uma parte experimental. Os argumentos advindos das análises dos dados estão fundamentados na Teoria Antropológica do Didático. Trabalhamos com a elaboração de recursos educacionais digitais para o ensino da Trigonometria utilizando o software GeoGebra, que serão aplicados a um grupo de alunos do Ensino Médio. Relacionada à pesquisa está a elaboração de um produto educacional que reunirá as tarefas aplicadas com uma orientação para a sua utilização em sala de aula.
We present a research which investigates the GeoGebra software capabilities as technological instrument conducive to learning of the trigonometric functions in high school, from the graphical representations of these functions. It is a survey characterized by a qualitative approach in which a didactic sequence of activities was used, with tasks at a level of growing demand. Each of this research execution phase is supported by the theoretical assumptions of the Didactic Engineering methodology used in research involving an experimental part. The arguments arising from the analysis are based on the Anthropological Theory of Didactic. For the development of digital educational resources for teaching trigonometry, which will apply to a group of high school students, used the GeoGebra software. Related to research, educational product offered here brings together the applied tasks and with guidance for using it in the classroom.

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Rosenbaum, Luciane Santos. "Uma trajetória hipotética de aprendizagem sobre funções trigonométricas numa perspectiva construtivista." Pontifícia Universidade Católica de São Paulo, 2010. https://tede2.pucsp.br/handle/handle/10826.

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Secretaria da Educação do Estado de São Paulo
This present work aims to verify: as compatible constructivist perspectives of learning with the planning of Trigonometric Functions; teaching the as researches in mathematics education field , which brings important results on the learning process may contribute to the organization of the Trigonometric Functions teaching that leverage best learning situations for students, as the performance of teachers of mathematics is revealed, with regard to planning activities in the teaching of Trigonometric Functions, consistent with a constructivist view of learning. We developed a qualitative study with two teachers and 70 students from the 2nd Grade of high school in a public school in the State of São Paulo. Its theoretical work of Simon (1995) on the use of HLT in teaching mathematics to formulate models of teaching based on constructivism. As a component of the Mathematics Teaching Cycle developed by Simon, the elaborate HLT was made use of the research findings for the development of Trigonometric Functions through activities and solve problems involving: constructions with ruler and compass, manipulative material, scientific calculator, construct graphs using software GeoGebra and paper and pencil. The results led us to conclude that the use of research contributes to the education organization of Trigonometric Functions; however you must provide access to such teachers to such research. Although the HLT are potentially rich, complex is the task of developing activities to accomplish a constructivist learning perspective. We note that participation in tasks involving the use of technology and material handling enhances the learning of Trigonometric Functions. However, the HLT is not prepared enough for learning to occur, because the teacher performance has a decisive role in mediating the construction of knowledge of the students. In the same way we experience the interaction and participation between students and teacher which is essential for learning
O presente trabalho tem como objetivo verificar: como compatibilizar perspectivas construtivistas de aprendizagem com o planejamento do ensino de Funções Trigonométricas; como as pesquisas na área de Educação Matemática, que trazem resultados importantes sobre a aprendizagem, podem contribuir para a organização do ensino de Funções Trigonométricas que potencialize boas situações de aprendizagem aos alunos; como a atuação do professor de Matemática se revela, no que se refere às atividades de planejamento do ensino de Funções Trigonométricas, de forma compatível com uma perspectiva construtivista de aprendizagem. Desenvolvemos um estudo de natureza qualitativa com 2 professores e 70 alunos da 2.ª série do Ensino Médio de uma escola da rede pública do Estado de São Paulo. Este trabalho, tem como fundamentação teórica os trabalhos de Simon (1995) sobre o uso de THA no ensino de Matemática para formular modelos de ensino baseados no construtivismo. Como componente do Ciclo de Ensino de Matemática desenvolvido por Simon, a THA elaborada fez uso de resultados de pesquisas para o desenvolvimento de Funções Trigonométricas por meio de atividades e resolução de problemas que envolveram: construções com régua e compasso, material manipulativo, calculadora científica, construção de gráficos usando o software Geogebra e papel e lápis. Os resultados obtidos nos levaram a concluir que o uso de pesquisas contribui para a organização do ensino de Funções Trigonométricas, no entanto é necessário possibilitar o acesso dos professores a tais pesquisas. Verificou-se que embora as THAs sejam potencialmente ricas, é complexa a tarefa de elaboração de atividades para que se efetive uma aprendizagem numa perspectiva construtivista. Constatamos que a participação em tarefas que envolvem o uso de tecnologia e manipulação de materiais potencializa o aprendizado de Funções Trigonométricas. Porém, a THA elaborada não é suficiente para que a aprendizagem ocorra, pois a atuação do professor tem papel decisivo na mediação da construção do conhecimento dos seus alunos. Da mesma forma vimos que a interação entre alunos, e estes com o professor são essenciais para uma aprendizagem significativa

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Yamamoto, Shuichi, and Naonori Ishii. "A way of computer use in mathematics teaching -The effectiveness that visualization brings-." Saechsische Landesbibliothek- Staats- und Universitaetsbibliothek Dresden, 2012. http://nbn-resolving.de/urn:nbn:de:bsz:14-qucosa-81179.

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We report a class of the mathematics in which an animation technology (calculating and plotting capabilities) of the software Mathematica is utilized. This class is taught for university students in a computer laboratory during a second semester. It is our purpose to make a student realize the usefulness and the importance of mathematics easily through visualization. In addition, we hope that students will acquire a new power of mathematics needed in the 21st century. For several years, we have continued this kind of class, and have continued to investigate the effectiveness that our teaching method (especially visualization) brings in the understanding of the mathematics. In this paper, we present some of this teaching method, which is performed in our class. From the questionnaire survey, it is found that our teaching method not only convinces students that the mathematics is useful or important but also deepens the mathematic understanding of students more.

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Crisafulli, Daniela. "Advanced modelling of multilayered composites and functionally graded structures by means of Unified Formulation." Thesis, Paris 10, 2013. http://www.theses.fr/2013PA100055/document.

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La plupart des problèmes d'ingénierie des deux derniers siècles ont été résolus grâce à des modèles structuraux pour poutres, plaques et coques. Les théories classiques, tels que Euler-Bernoulli, Navier et de Saint-Venant pour les poutres, et Kirchhoff-Love et Mindlin-Reissner pour plaques et coques, ont permis de réduire le problème générique 3-D, dans le problème unidimensionnel pour les poutres et deux dimensionnelle pour les coques et les plaques. Théories raffinés d'ordre supérieur ont été proposées au cours du temps, comme les modèles classiques ne consentez pas à d'obtenir une complète domaine des contraintes et des déformations. La Carrera Unified Formulation (UF) a été proposé au cours de la dernière décennie, et permet de développer un grand nombre de théories structurelles avec un nombre variable d'inconnues principales au moyen d'une notation compacte et se référant à des nuclei fondamentales. Cette formulation unifiée permet de dériver carrément des modèles structurels d'ordre supérieur, pour les poutres, plaques et coques. Dans ce cadre, cette thèse vise à étendre la formulation pour l'analyse des structures fonctionnellement gradués (FGM), en introduisant aussi le problème thermo-mécanique, dans le cas des poutres fonctionnellement gradués. Suite à la formulation unifiée, les variables génériques déplacements sont écrits en termes de fonctions de base, qui multiplie les inconnues. Dans la deuxième partie de la thèse, de nouvelles fonctions de bases pour la modélisation des coques, qui représentent une approximation trigonométrique des variables déplacements, sont pris en compte
Most of the engineering problems of the last two centuries have been solved thanks to structural models for both beams, and for plates and shells. Classical theories, such as Euler-Bernoulli, Navier and De Saint-Venant for beams, and Kirchhoff-Love and Mindlin- Reissner for plates and shells, permitted to reduce the generic 3-D problem, in onedimensional one for beams and two-dimensional for shells and plates. Refined higher order theories have been proposed in the course of time, as the classical models do not consent to obtain a complete stress/strain field. Carrera Unified Formulation (UF) has been proposed during the last decade, and allows to develop a large number of structural theories with a variable number of main unknowns by means of a compact notation and referring to few fundamental nuclei. This Unified Formulation allows to derive straightforwardly higher-order structural models, for beams, plates and shells. In this framework, this thesis aims to extend the formulation for the analysis of Functionally Graded structures, introducing also the thermo-mechanical problem, in the case of functionally graded beams. Following the Unified Formulation, the generic displacements variables are written in terms of a base functions, which multiplies the unknowns. In the second part of the thesis, new bases functions for shells modelling, accounting for trigonometric approximation of the displacements variables, are considered

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42

Saini, Laura. "Nouveaux outils pour l'animation et le design : système d'animation de caméra pour la stop motion, fondée sur une interface haptique et design de courbes par des courbes algébriques-trigonométriques à hodographe pythagorien." Phd thesis, Université de Valenciennes et du Hainaut-Cambresis, 2013. http://tel.archives-ouvertes.fr/tel-00835671.

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Dans la première partie de la thèse, nous présentons un nouveau système permettant de produire des mouvements de caméra réalistes pour l'animation stopmotion. Le système permettra d'enrichir les logiciels d'animation 3D classiques (comme par exemple Maya et 3D Studio Max) afin de leur faire contrôler des mouvements de caméra pour la stop motion, grâce à l'utilisation d'une interface haptique. Nous décrivons le fonctionnement global du système. La première étapeconsiste à récupérer et enregistrer les données envoyées par le périphérique haptique de motion capture. Dans la seconde étape, nous réélaborons ces données par un procédé mathématique, puis les exportons vers un logiciel de 3D pour prévisualiser les mouvements de la caméra. Finalement la séquence est exécutée avec un robot de contrôle de mouvement et un appareil photo. Le système est évalué par un groupe d'étudiants du Master "Art plastiques et Création numérique" de l'Université de Valenciennes. Dans la deuxième partie, nous définissons une nouvelle classe de courbes à partir des courbes polynomiales paramétriques à hodographe pythagorien (PH) construite sur un espace algébrique-trigonométrique. Nous montrons leurs propriétés fondamentaleset leurs avantages importants par rapport à leur équivalent polynomial, grâce à l'utilisation d'un paramètre de forme. Nous introduisons une formulation complexe et nous résolvons le problème d'interpolation de Hermite.

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Sipos, Maksim. "Dynamical plane structures in the parameter plane of cosine-root family." Thesis, 2007. http://www.ithaca.edu/hs/depts/math/docs/theses/siposthesis.pdf.

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Mosese, Nthabiseng Mamotho. "Evaluating the effectiveness of the use of information and communication technology in the teaching and learning of trigonometry functions in grade 12." Diss., 2017. http://hdl.handle.net/10500/23449.

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The high school pass rate, for mathematics, in South Africa is very low. This is particularly so in trigonometry functions. One of the possible factors leading to this is the traditional method of teaching and learning. This study was undertaken to determine whether the use of Information and Communication Technology (ICT) would influence students’ learning of trigonometry functions. In order to answer this question the teaching and learning instructions developed were based on activity theory (AT) and action, process, object, and schema (APOS) theory. The study followed a non-equivalent control group, quasi-experimental design with a pre- post-test approach. Since it was not possible to randomly select participants for the study, intact groups were used. There were two groups: a control and an experimental one. Both groups wrote a standardized achievement pre-test to establish their comparability at the beginning of the study. While the control group was taught in the traditional way (grade 10-12 syllabus), the experimental group used the software Geogebra. The computer software (Geogebra) and the South African grade 10-12 syllabus for trigonometry functions were used during the lessons of the experimental group. At the end of the study, a similar post-test was administered on both groups to measure the comparative effects of either of the teaching methods on the performance of students. A t-test independent sample statistical analysis was performed on the findings using a statistics package, SPSS. The results of this investigation indicated that the use of the computer software, Geogebra, in the teaching and learning of trigonometry functions improved the performance of the Grade 12 students.
Mathematics Education
M. Sc. (Mathematics Education)

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Fransman, Johanna Sandra. "Mathematics teachers' metacognitive skills and mathematical language in the teaching-learning of trigonometric functions in township schools / Johanna Sandra Fransman." Thesis, 2014. http://hdl.handle.net/10394/13942.

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Metacognition is commonly understood in the context of the learners and not their teachers. Extant literature focusing on how Mathematics teachers apply their metacognitive skills in the classroom, clearly distinguishes between teaching with metacognition (TwM) referring to teachers thinking about their own thinking and teaching for metacognition (TfM) which refers to teachers creating opportunities for learners to reflect on their thinking. However, in both of these cases, thinking requires a language, in particular appropriate mathematical language to communicate the thinking by both teacher and learners in the Mathematics classroom. In this qualitative study, which forms part of a bigger project within SANPAD (South Africa Netherlands Research Programme on Alternatives in Development), the metacognitive skills and mathematical language used by Mathematics teachers who teach at two township schools were interrogated using the design-based research approach with lesson study. Data collection instruments included individual interviews and a trigonometric assessment task. Lessons were also observed and video-taped to be viewed and discussed during focus group discussions in which the teachers, together with five Mathematics lecturers, participated. The merging of the design-based research approach with lesson study brought about teacher-lecturer collaboration, referred to in this study as the Mathematics Educators’ Reflective Inquiry (ME’RI) group, and enabled the design of a hypothetical teaching and learning trajectory (HTLT) for the teaching of trigonometric functions. A metacognitive performance profile for the two grade 10 teachers was also developed. The Framework for Analysing Mathematics Teaching for the Advancement of Metacognition (FAMTAM) from Ader (2013) and the Teacher Metacognitive Framework (TMF) from Artzt and Armour-Thomas (2002) were adjusted and merged to develop a new framework, the Metacognitive Teaching for Metacognition Framework (MTMF) to analyse the metacognitive skills used by mathematics teachers TwM as well as TfM. Without oversimplifying the magnitude of these concepts, the findings suggest a simple mathematical equation: metacognitive skills + enhanced mathematical language = conceptualization skills. The findings also suggest that both TwM and TfM are required for effective mathematics instruction. Lastly the findings suggest that the ME’RI group holds promise to enhance the use of the metacognitive skills and mathematical language of Mathematics teachers in Mathematics classrooms.
PhD (Mathematics Education), North-West University, Potchefstroom Campus, 2014

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Chen, Hui-yu, and 陳惠瑜. "On generalized trigonometric functions." Thesis, 2010. http://ndltd.ncl.edu.tw/handle/47793825586318881299.

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碩士
國立中山大學
應用數學系研究所
98
The function $sin x$ as one of the six trigonometric functions is fundamental in nearly every branch of mathematics, and its applications. In this thesis, we study an integral equation related to that of $sin x$: $mbox{~for~}xin[-frac{hat{pi}_{p}}{2},~frac{hat{pi}_{p}}{2}] mbox{~and~} p>1$ $$x=int_0^{S_{p}(x)}(1-|t|^{p})^{-frac{1}{p}}dt.$$ Here $hat{pi}_{p}=frac{2pi}{psin(frac{pi}{p})}=2int_0^1(1-t^{p})^{-frac{1}{p}}dt.$ We find that the function $S_{p}(x)$ is well defined. Its properties are also similar to those of $sin x$ : differentiation, identities, periodicity, asymptotic expansions, $cdots$, etc. For example, we have $$|S_{p}(x)|^{p}+|S''_{p}(x)|^{p}=1mbox{~~and~~}frac{d}{dx}(|S''_{p}(x)|^{p-2}S''_{p}(x))=-(p-1)|S_{p}(x)|^{p-2}S_{p}(x).$$ We call $S_{p}(x)$ the generalized sine function. Similarly, we define the generalized cosine function $C_{p}(x)$ by $|x|=int_{C_{p}(x)}^{1}(1- t^{p})^{-frac{1}{p}}dt$ for $xin[-frac{hat{pi}_{p}}{2}$,~$frac{hat{pi}_{p}}{2}]$ and derive its properties. Thus we obtain two sets of trigonometric functions: egin{itemize} item[(i)]$~S_{p}(x),~ S''_{p}(x),~ T_{p}(x)=frac{S_{p}(x)}{S''_{p}(x)},~RT_{p}(x)=frac{S''_{p}(x)}{S_{p}(x)},~ SE_{p}(x)=frac{1}{S''_{p}(x)},~ RS_{p}(x)=frac{1}{S_{p}(x)}~;$ item[(ii)]$~C_{p}(x),~ C''_{p}(x),~RCT_{p}(x)=-frac{C''_{p}(x)}{C_{p}(x)},~ CT_{p}(x)=-frac{C_{p}(x)}{C''_{p}(x)},~RC_{p}(x)=frac{1}{C_{p}(x)},~ CS_{p}(x)=-frac{1}{C''_{p}(x)}mbox{~。~}$ end{itemize}These two sets of functions have similar differentiation formulas, identities and periodic properties as the classical trigonometric functions. They coincide when $p=2$. Their graphs and asymptotic expansions are also interesting. Through this study, we understand more about the theoretical framework of trigonometric functions.

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WU, CHIA-SUNG, and 吳家松. "The Capability Index of Learning the Trigonometric Function:A Case Study of Sinusoidal Function." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/gdcrte.

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碩士
國立臺北教育大學
數位科技設計學系(含玩具與遊戲設計碩士班)
104
The purpose of this thesis was to describe the teaching trigonometric functions, which could help the learning effect in the course. In the beginning, the trigonometric history was to understand the trigonometric English and Chinese meaning or definition in the trigonometric teaching and learning curriculum, which could inspire the concept of distance measurement. Secondly, the notion of the wave was stimulated in the sinusoidal curve. The properties of wave implied the life and power energy, it was a pleasure to learn in the trigonometric functions. In other words, the characterization of sin function had not only the measurement distance represented but also the life and signal. Then, the GSP computer graphics software was employed in the trigonometric teaching, which could obtain the enjoyment of moving graphics software. Teaching not only to teach trigonometric calculation, measurement distance method, develop logical thinking, and, more importantly, led fluctuation in the world, inspire innovation, combined with GSP computer graphics software, the ability to build scientific capability. Hope for the future of learning, to stimulate creativity. Finally, the mathematical modeling was introduced in in the trigonometric curriculum plan, which could establish a scientific attitude and spirit in the learning procedure. Based on the author's own teaching and working experience, the writer finished the teaching trigonometric table, which included the new wave concept and procedure in the teaching course. The object of this study was to obtain the win-win effect for the teacher and student each other, which could enjoy the resonance effect in the teaching and learning of the trigonometric function.

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Yang, Yu-wen, and 楊喻文. "Topics on Generalized Trigonometric And Hyperbolic Functions." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/81066819521511024824.

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碩士
國立中山大學
應用數學系研究所
101
This thesis is a continuation of the master thesis of Hui-Yu Chen in 2009. We study the generalized sine functions Sp and generalized cosine functions Cp in more detail. In particular, we evaluate the definite integrals of ∫_0^(πp/2) Sp (x)^α S''p (x)^β dx and ∫_0^(πp/2) C_p (x)^α |C''p (x)|^β dx α> -1 and β>1-p in terms of gamma function. Following the work of Binding et al [2], we also give a proof that the sequence {Sp (nπp x)} form a Riesz basis in L^2(0,1) and a Schauder basis in L^q(0,1) for ( q>6/5 ) . On the other hand, we define the generalized hyperbolic sine function Shp and generalized hyperbolic cosine function Chp as x=∫_0^(Shp (x))(1+|t|^p)^(-1/p) dt and |x|=∫_1^Chp (x)(t^p-1)^(-1/p) dt When p=2, they become the hyperbolic sine and cosine functions. We show that the definition is equivalent to the identity and the associated half-linear equation for each function. Furthermore we evaluate the improper integrals ∫_0^∞ Sh_p (x)^α Sh^''p (x)^β dx 和 ∫_0^∞ Ch_p (x)^α Ch''_p (x)^β dx ,α>-1,α+β<0 and β>1-p,α+β<0, again in terms of gamma functions.

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Wu, Jyun-Sian, and 吳俊賢. "Trigonometric functions with Chaotic for image scrambling." Thesis, 2013. http://ndltd.ncl.edu.tw/handle/66822111734061671749.

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碩士
聖約翰科技大學
自動化及機電整合研究所
101
Abstract This thesis proposes an image scrambling method based on trigonometric functions to produce a chaotic sequence. It includes 1. Design chaotic sequence equations based on trigonometric functions. 2. Simulations and analysis for the proposed method applying to black and white images, gray-level images and color images. 3. Compare with Logistic chaotic methods, Arnold transformation methods by image scrambling degrees, and robustness for occlusion attack and noise attack. 4. Sensitivity analysis of initial values for the proposed method. 5. Apply to inequilaterally-length images. Keywords: Image Scrambling, Chaotic, Trigonometric functions.

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HUANG, JHIN-JHUN, and 黃智駿. "Topics on Generalized (p,q)-Trigonometric Function." Thesis, 2016. http://ndltd.ncl.edu.tw/handle/5gub2k.

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碩士
國立臺北教育大學
數學暨資訊教育學系(含數學教育碩士班)
104
In this thesis, we firstly discuss the monotonicity of S_{p}(x) and pi_{p} in p. Here S_{p} is the inverse function of the integral x=\int^{S_{p}(x)}_{0}(1-|t|^{p})^{-\frac{1}{p}}dt, p>1 and pi_{p}=2\int^{1}_{0}(1-t^{p})^{-\frac{1}{p}}dt. We revise the mistake that Binding made ,and prove the function S_{p} and pi_{p} decrease in p. In the second part, we extend the definitions of S_{p} and pi_{p} to S_{p,q} and pi_{p,q}, and discuss the relative differential equations and the monotonicity in p and q. Here S_{p,q} is the inverse function of the integral x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt ,p,q>1 and pi_{p,q}=2\int^{1}_{0}(1-t^{q})^{-\frac{1}{p}}dt. We find that S_{p,q}, S_{p} and sin(x) have similar property, and prove the following equivalent conditions \begin{enumerate} (a)\forall x\in[-\frac{\pi_{p,q}}{2},\frac{\pi_{p,q}}{2}] x=\int^{S_{p,q}(x)}_{0}(1-|t|^{q})^{-\frac{1}{p}}dt.\forall x \in \mathbb{R} S_{p,q}(x+k\pi_{p,q})=(-1)^{k}S_{p,q}(x). (b)S_{p,q}(0)=0,S'_{p,q}(0)=1 |S_{p,q}|^{q}+|S'_{p,q}|^{p}=1 (c)S_{p,q}(0)=0,S'_{p,q}(0)=1 (|S'_{p,q}|^{p-2}S'_{p,q})'+\frac{q(p-1)}{p}|S_{p,q}|^{q-2}S_{p,q}=0 Finally, we also prove that pi_{p,q} and S_{p,q}(\pi_{p,q}t),\forall t\in[0,1] are decreasing in p,q.

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Dissertations / Theses on the topic 'Trigonometry functions'

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