10  Copernican Trigonometry pp. 7388By Victor J. Katz
Mathematical Time Capsules
MAA Notes
(No. 77)
Online ISBN: 9780883859841
Chapter DOI: http://dx.doi.org/10.5948/UPO9780883859841.011
Subjects: Recreational Mathematics, History of Science and Technology 

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Introduction
In most trigonometry courses, the instructor begins by defining the sine, cosine, and tangent of an angle as ratios of certain sides in an appropriate right triangle. She then proceeds to calculate, using elementary geometry, the sine, cosine and tangent of angles of 30°, 45°, and 60°. But once students need to calculate the sine of 27°, they are told to punch some buttons on their calculators. What do students think happens when they do that? Do they imagine that somewhere inside the calculator, someone draws a miniature right triangle with one base angle 27°, then measures the sides and divides? Where do these numbers come from that so miraculously appear on the calculator screen in half a second?
Fifty years ago, no one had calculators. Then, the trigonometry texts simply told the students to consult the table at the back of the book to find the sine of 27°. That took a bit longer, but still, there was little in the text to show students where those numbers came from. They just “were”. Whether one uses tables or uses calculators, it still seems that there is a mystery in these numbers that should not exist. Most teachers certainly want their students to be fluent in calculator use – and these are generally easier to use than tables. But still, we do not want students thinking that calculators are magic.