In this project, we explore the genesis of the trigonometric functions: sine, cosine, tangent, cotangent, secant, and cosecant. The goal is to provide the typical student in a pre-calculus course some context for understanding these concepts that is generally missing from standard textbook developments. Trigonometry emerged in the ancient Greek world (and, it is suspected, independently in China and India as well) from the geometrical analyses needed to solve basic astronomical problems regarding the relative positions and motions of celestial objects. While the Greeks (Hipparchus, Ptolemy) recognized the usefulness of tabulating chords of central angles in a circle as aids to solving problems of spherical geometry, Hindu mathematicians, like Varahamahira (505--587), found it more expedient to tabulate half-chords, whence the use of the sine and cosine became popular. We will examine an excerpt from this work, wherein Varahamahira describes a few of the standard modern relationships between sine and cosine in the course of creating a sine table. In the 11th century, the Arabic scholar and expert on Hindu science Abu l-Rayhan Muhammad al-Biruni (973--1055) published The Exhaustive Treatise on Shadows (ca.~1021). In this work, we see how Biruni presents geometrical methods for the use of sundials; the relations within right triangles made by the gnomon of a sundial and the shadow cast on its face lead to the study and tabulation of values of the tangent and cotangent, secant and cosecant. Biruni also works out the relationships that these quantities have with the sines and cosines of the angles. However, the modern terminology for the standard trigonometric quantities is not established until the European Renaissance. Foremost in this development is the landmark On Triangles (1463) by Regiomontanus (Johannes Muller). Regiomontanus exposes trigonometry in a purely geometrical form and then applies the ideas to problems in circular and spherical geometry. We examine a few of the theorems that explore the trigonometric relations and which are used to solve triangle problems.
Applicable Math Courses
Lower Division Undergraduate
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Otero, Danny, "A Genetic Context for Understanding the Trigonometric Functions" (2017). Pre-calculus and Trigonometry. 1.