For integrals consisting of rational functions of sine and cosine a set of little known rules known as the Bioche rules are considered. The rules, which consist of testing the differential form of the integral for invariance under one of three simple substitutions x [right arrow] -- x, [pi] -- x, and [pi] + x, allow one to decide which of the…

The aim of this study is to analyse undergraduate students' understanding of polar coordinates based on two theories, Action, Process, Object and Schema (APOS) and Onto-Semiotic Approach (OSA). These two theories complement each other and each of them separately has been used in many research to explore students' performance of mathematical…

In this paper, we highlight examples from school mathematics in which invariance did not receive the attention it deserves. We describe how problems related to invariance stimulated the interest of both teachers and students. In school mathematics, invariance is of particular relevance in teaching and learning geometry. When permitted change…

Including opportunities for students to experience uncertainty in solving mathematical tasks can prompt learners to resolve the uncertainty, leading to mathematical understanding. In this article, we examine how preservice secondary mathematics teachers' thinking about a trigonometric relationship was impacted by a series of tasks that prompted…

A geometrical task is presented with multiple solutions using different methods, in order to show the connection between various branches of mathematics and to highlight the importance of providing the students with an extensive 'mathematical toolbox'. Investigation of the property that appears in the task was carried out using a computerized tool.

Covariational reasoning has been the focus of many studies but only a few looked into this reasoning in the polar coordinate system. In fact, research on student's familiarity with polar coordinates and graphing in the polar coordinate system is scarce. This paper examines the challenges that students face when plotting polar curves using the…

Smartphones used as tools provide opportunities for the teaching of the concepts of accuracy and precision and the mathematical concept of arctan. The accuracy and precision of a trigonometric experiment using entirely mechanical tools is compared to one using electronic tools, such as a smartphone clinometer application and a laser pointer. This…

In computing real-valued functions, it is ordinarily assumed that the input to the function is known, and it is the output that we need to approximate. In this work, we take the opposite approach: we show how to compute the values of some transcendental functions by approximating the input to these functions, and obtaining exact answers for their…

This article focuses on a simple trigonometric problem that generates a strange phenomenon when different methods are applied to tackling it. A series of problem-solving activities are discussed, so that students can be alerted that the precision of diagrams is important when solving geometric problems. In addition, the problem-solving plan was…

In this short note, by using one of Li and Liu's theorems [K.-H. Li, "The solution of CIQ. 39," "Commun. Stud. Inequal." 11(1) (2004), p. 162 (in Chinese)], "s-R-r" method, Cauchy's inequality and the theory of convex function, we solve some geometric inequalities conjectures relating to an interior point in triangle. (Contains 1 figure.)

A closed form solution for the trigonometric integral [integral]sec[superscript 2k+1]xdx, k=0,1,2,..., is presented in this article. The result will fill the gap in another trigonometric integral [integral]sec[superscript 2m+1] x tan[superscript 2n]xdx, which is neglected by most of the calculus textbooks due to its foreseeable unorthodox solution…

A generalization of a well-known result for the arctangent function poses a number of interesting questions concerning the existence of integer solutions of related problems.

We discuss a class of trigonometric functions whose corresponding Fourier series, on a conveniently chosen interval, can be used to calculate several numerical series. Particular cases are presented and two recent results involving numerical series are recovered. (Contains 1 note.)

A complex technology-based problem in visualization and computation for students in calculus is presented. Strategies are shown for its solution and the opportunities for students to put together sequences of concepts and skills to build for success are highlighted. The problem itself involves placing an object under water in order to actually see…

Non-linear second-order differential equations whose solutions are the elliptic functions "sn"("t, k"), "cn"("t, k") and "dn"("t, k") are investigated. Using "Mathematica", high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are…