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. 1) Did you guess correctly? 2) What do you notice about how Sketchpad measures angles greater than pi? 3) What properties does this locus have? Can you explain them? 4) Consider & explore other variations for D (including using other regular polygons).

Square Trigonometry

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Created by Michael de Villiers, 27 June 2010; updated 15 June 2019.