# ma007 - Sines Cosines of the Times Victor J Katz Math...

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Victor J. Katz Math Horizons, April 1995, p. 5. Victor J. Katz, visiting mathematician at the MAA during the 1994–95 academic year, teaches at the University of the District of Columbia. His textbook, A History of Mathematics: An Introduction , was published by HarperCollins in 1993. W hy does the derivative of the sine equal the cosine? Or the derivative of the tangent equal the square of the secant? One answer, that you learned early in your calculus course, is that these rules can be proved. In fact, your instructor probably proved the first from the definition of derivative, having first convinced you that and proved the second by using the quotient rule. But, after all, the trigonometric functions are defined geometrically; one ought to be able to understand their derivatives geometrically as well. If we look back at the history of these functions and their relationship to the history of calculus, we can do exactly that. Today, we generally consider the sine and the other trigonometric functions as numerical functions of real numbers, where the numbers in the domain can be thought of as measures of angles. But until the time of Euler in the mid-eighteenth century, sines were

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ma007 - Sines Cosines of the Times Victor J Katz Math...

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