## Square Trigonometry

The trigonometric ratios are usually defined as functions at school level in terms of a unit circle. But *what happens if*, as intriguingly suggested by Johan Gielis from Belgium in a talk at the Croatian Mathematical Conference in June 2010, instead of the usual circle, we move a point around a different regular figure, say a square?

In the sketch below, the angle *ABC* has been measured in radians, as well as the height *y* of the point *C* as it moves around the circumference of the square. The point *D* was then plotted with the coordinates (angle *ABC* + *pi*, *y*). Can you guess BEFOREHAND the locus of *D*?

Investigate the locus of *D* by first dragging point *C* around the perimeter, and then clicking on the locus button. Did you guess correctly? What do you notice about how *Sketchpad* measures angles greater than *pi*? What properties does this locus have? Can you explain them?

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Square Trigonometry

This page uses **JavaSketchpad**, a World-Wide-Web component of *The Geometer's Sketchpad.* Copyright © 1990-2008 by KCP Technologies, Inc. Licensed only for non-commercial use.

Michael de Villiers, 27 July 2010.