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 discrete Fourier series is a valuable tool developed and used by mathematicians and engineers alike. One of the most prominent applications is signal processing. Usually, it is important that the signals be transmitted fast, for example, when transmitting images over large distances such as between the moon and the earth or when generating…

Pythagoras Theorem is an old mathematical treatise that has traversed the school curricula from secondary to tertiary levels. The patterns it produced are quite interesting that many researchers have tried to generate a kind of predictive approach to identifying triples. Two attempts, namely Diophantine equation and Brahmagupta trapezium presented…

Finding patterns and making conjectures are important thinking skills for students at all levels of mathematics education. Both the Common Core State Standards for Mathematics and the National Council of Teachers of Mathematics speak to the importance of these thought processes. NCTM suggests that students should be able to recognize reasoning and…

Pythagoras' theorem in two and three dimensions appears in General Mathematics, Units 1-2, section 6 (Geometry and trigonometry: Shape and measurement) in the Victorian Certificate of Education Mathematics Study Design (Victorian Curriculum Assessment Authority, 2010). It also comes in Further Mathematics, Units 3-4 (Applications: Geometry and…

This note gives an alternative formulation of Euler's formula.

We discuss a general class of trigonometric functions whose corresponding Fourier series can be used to calculate several interesting numerical series. Particular cases are presented. (Contains 4 notes.)

In the present note, two Parseval-type relations involving the Laplace transform are given. The application of the relations is demonstrated in evaluating improper integrals and Laplace transforms of trigonometric functions.

In the year 1837 mathematical proof was set forth authoritatively stating that it is impossible to trisect an arbitrary angle with a compass and an unmarked straightedge in the classical sense. The famous proof depends on an incompatible cubic equation having the cosine of an angle of 60 and the cube of the cosine of one-third of an angle of 60 as…