This report discusses the background of trigonometric sequences and series related to defining the sine and cosine functions. Proofs involving converging trigonometric sequences and series are presented using nontraditional methods. To conclude, an application of trigonometric sequences and series is shown. / text

This thesis is intended as an introduction to the study of one type of trigonometric series, the Fourier series.

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No description available.

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.

This thesis is a study of the analytical development of the trigonometric functions.

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.

<p>This work is devoted to trigonometric series with monotone coefficients. The main problem is to study conditions under which a given series is the Fourier series of an integrable (or continuous) function.</p>

This work is devoted to trigonometric series with monotone coefficients. The main problem is to study conditions under which a given series is the Fourier series of an integrable (or continuous) function.

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel Superior / O presente trabalho foi dividido em trÃs partes: trigonometria no triÃngulo retÃngulo, trigonometria no ciclo trigonomÃtrico, nÃmeros complexos. No triÃngulo retÃngulo foram definidos os valores do seno, cosseno, tangente, cotangente, cossecante e secante dos Ãngulos notÃveis: 18Â, 30Â, 45Â, 60Â alÃm das suas derivaÃÃes. Propriedades importantes como a relaÃÃo trigonomÃtrica fundamental foram demonstradas. No ciclo trigonomÃtrico alÃm das propriedades advindas do triÃngulo retÃngulo foram apresentadas e provadas outras como as leis do seno e do cosseno, relaÃÃes trigonomÃtricas de Ãngulos maiores que 90Â e da soma e diferenÃa de arcos, equaÃÃes trigonomÃtricas. Na parte de nÃmeros complexos foi apresentado o nÃmero i e suas propriedades juntamente com as formas algÃbrica e geomÃtrica de um nÃmero complexo. Neste ponto foi visto a importÃncia da trigonometria para o desenvolvimento da fÃrmula de Moivre. No apÃndice, temos provado as potÃncias do nÃmero (i) e a tabela trigonomÃtrica. / This study was divided into three parts: the right triangle trigonometry, trigonometry in trigonometric cycle, complex numbers. In the right triangle the sine values were defined, cosine, tangent, cotangent, cosecant and drying of the remarkable angles: 18Â, 30Â, 45Â, 60Â beyond its derivations. Important properties as the fundamental trigonometric relationship were demonstrated. Trigonometric cycle in addition to the resulting properties of the right triangle were presented and other proven as the laws of sine and cosine, trigonometric relationship of angles greater then 90Â and the sum and difference of arcs, trigonometric equations. In the complex numbers was made the number in their properties along with the algebraic and geometric forms a complex number. At this point it has been seen trigonometric to the importance of the development of Moivre formula. In the appendix we have tasted the powers of the number (i) and the trigonometric table.