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…

This article presents a task providing college students opportunities to build on their high school knowledge of trigonometry to explore parametric equations and inverse trigonometric relationships within a contextual learning ladder problem.

This article describes how mathematics may be experienced in widely different ways across mathematics and physics courses and highlights some unexpected constraints in a trigonometry curriculum. The examples and discussion are based on a study (Mateas 2020) that compares how trigonometry is portrayed in representative physics (i.e., "Holt…

In the late 1800s and early 1900s, increases in metallurgic technology and better manufacturing methods made naval artillery a more powerful force. Guns could fire more powerful shells that could travel farther and hit a target with much greater accuracy. Torpedoes represented a major threat to even the most powerful of warships, forcing captains…

In this article, the author presents a Desmos activity where students adjust the measures of angles in radians to reposition a laser and a mirror so the beam passes through three stationary targets. This Radian Lasers activity can be extended to simulate project-based learning (PBL), a pedagogical approach for applying concepts and skills from…

In high school, students extend understanding of linear and exponential functions and explore trigonometric functions. This includes using the unit circle to connect trigonometric functions to their geometric foundation, modeling periodic phenomena, and applying (and proving) trigonometric identities. These ideas are fundamental for trigonometric…

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…

In mathematics, there are countless connections between apparently unrelated topics. Students often find these relationships interesting, and they contribute to expanding their mathematical repertoires. This article describes one of these unusual connections, between the sum of periodic functions and the commensurable and incommensurable numbers.…

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°.

If students are asked, "What is the world water crisis?" how would they respond? In the authors precalculus classes, this question is often met with blank stares and students avoiding eye contact. Many students are unaware of the world water crisis--what it is, why it is, or who it affects. Even fewer realize the utility of trigonometry…

High school students often ask questions about the nature of infinity. When contemplating what the "largest number" is, or discussing the speed of light, students bring their own ideas about infinity and asymptotes into the conversation. These are popular ideas, but formal ideas about the nature of mathematical sets, or "set…

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…

Historical accounts of trigonometry refer to the works of many Indian and Arab astronomers on the origin of the trigonometric functions as we know them now, in particular Abu al-Wafa (ca. 980 CE), who determined and named all known trigonometric functions from segments constructed on a regular circle and later on a unit circle (Moussa 2011;…

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

A visual proof that tan x/x is monotonically increasing on (0, pi/2). For sin x/x, see p. 408 (EJ1017684).