For a given rational number r, a classical theorem of Niven asserts that if cos(rp) is rational, then cos(rp) [element-of] {0,±1,±1/2}. In this note, we extend Niven's theorem to quadratic irrationalities and present an elementary proof of that.

The mode of assessment is one of the most important factors influencing learning. E-assessment usually includes only checking the final answer, thus limiting teacher's ability to check the complete solution, and it does not allow inclusion of math proofs problems that constitute an important part of math content. The e-assessment module of Halomda…

In this paper, Richard Ollerton presents two alternative approaches to proving the six standard trigonometric angle sum and difference identities. They are suitable for students who have an understanding of rotation matrices.

Proof ability of prospective teachers on Trigonometry material is still lacking. It can be seen when they carry out trigonometry proof that does not meet the proof indicators to conduct the research. This study aimed at improving proof ability of prospective teachers with a contextual model on Trigonometry materials. The research method used was…

Modern calculus textbooks carefully illustrate how to perform integration by trigonometric substitution. Unfortunately, most of these books do not adequately justify this powerful technique of integration. In this article, we present an accessible proof that establishes the validity of integration by trigonometric substitution. The proof offers…

The purpose of this paper is to follow the reasoning of high school students when asked to explain the standard trigonometric limit lim/[theta][right arrow] sin[theta]/[theta]. An observational study was conducted in four different phases in order to investigate if visualization, by means of an interactive technology environment (Geogebra), can…

The purpose of this note is to discuss how paper folding can be used to find the exact trigonometric ratios of the following four angles: 22.5°, 67.5°, 27°, and 63°.

Mathematical reasoning with algebraic and geometric representations is essential for success in upperdivision and graduate-level physics courses. Complex algebra requires student to fluently move between algebraic and geometric representations. By designing a task for middle-division physics students to translate a geometric representation to…

Review of the history of trigonometry content and pedagogy indicates the necessity and importance of trigonometry in the school curriculum (e.g., van Brummelen, 2009; van Sickel, 2011). For example, understanding trigonometric functions is a requirement for understanding some other areas of science, such as Newtonian physics, architecture,…

Two important pedagogical techniques for developing deeper mathematical understanding are to prove a given theorem in different ways and to unify the proofs of different theorems. Trigonometric angle sum and difference identities are introduced in Unit 2 of Specialist Mathematics in the Australian Curriculum (Australian Curriculum, Assessment and…

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…

We present results from a classroom-based intervention designed to help a class of grade 10 students (14-15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students' solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs.…

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.

Solution of problems in mathematics, and in particular in the field of Euclidean geometry is in many senses a form of artisanship that can be developed so that in certain cases brief and unexpected solutions may be obtained, which would bring out aesthetically pleasing mathematical traits. We present four geometric tasks for which different proofs…

A visual proof that sin x/x is monotonically increasing on (0, pi/2). For tan x/x, see p. 420 (EJ1017686).