An original approach using three appropriate Pythagorean triangles is presented for the detailed mathematical analysis of an ideal conical pendulum. The triangles that are used in this analysis relate specifically to the physical dimensions of the conical pendulum, the magnitudes of the forces acting during the conical pendulum motion and a…

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…

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…

Trigonometry is a well known branch of Mathematics. The study of trigonometry is of great importance in surveying, astronomy, navigation, engineering, and in different branches of science. This paper reports on the discovery of flaws in the traditional definitions of trigonometric ratios of an angle, which (in most cases) make use of the most…

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…

Using a simple trigonometric limit, we provide an intuitive geometric proof of the Singular Value Decomposition of an arbitrary matrix.

One of the most rewarding accomplishments of working with preservice secondary school mathematics teachers is helping them develop conceptually connected knowledge and see mathematics as an integrated whole rather than isolated pieces. To help students see and use the connections among various mathematical topics, the authors have paid close…

In some secondary mathematics curricula, there is a topic called Stereometry that deals with investigating the position and finding the intersection, angle, and distance of lines and planes defined within a prism or pyramid. Coordinate system is not used. The metric tasks are solved using Pythagoras' theorem, trigonometric functions, and sine and…

Students from small rural schools (class 1A, 2A, and 3A) historically have not had access to dual credit courses for several reasons including distance from a college campus, affordability, and district teaching strength. In an effort to address these problems and to begin the college experience sooner, a new program was developed by the…

For almost all students, what happens when they push buttons on their calculators is essentially magic, and the techniques used are seemingly pure wizardry. In this article, the author draws back the curtain to expose some of the mathematics behind computational wizardry and introduces some fundamental ideas that are accessible to precalculus…

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.)

Using modern technology to examine classical mathematics problems at the high school level can reduce difficult computations and encourage generalizations. When teachers combine historical context with access to technology, they challenge advanced students to think deeply, spark interest in students whose primary interest is not mathematics, and…

Trigonometry is one of the topics in mathematics that the students in both high school and pre-undergraduate levels need to learn. Generally, the topic covers trigonometric functions, trigonometric equations, trigonometric identities and solving oblique triangles using the Laws of Sines and Cosines. However, when solving the oblique triangles,…

A portion of all students in introductory or developmental undergraduate mathematics courses find themselves at an unfortunate tipping point: the border between passing and failing. These high-stakes courses often come with high enrollments, and a recurring problem: high failure rates. The measure of success used at the author's institution, the…

This article presents a mathematical solution to a motorway problem. The motorway problem is an excellent application in optimisation. As it integrates the concepts of trigonometric functions and differentiation, the motorway problem can be used quite effectively as the basis for an assessment tool in senior secondary mathematics subjects.…