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# ma006 - The Spiders Spacewalk Derivation of sin and cos Tim...

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The Spider’s Spacewalk Derivation of sin and cos Tim Hesterberg, Franklin and Marshall College, Lancaster, PA 17604–3003 The College Mathematics Journal , March 1995, Volume 26, Number 2, pp. 144–145. T he usual proofs of the derivatives of sine and cosine in introductory calculus involve limits. I shall outline a simple geometric derivation that avoids evaluating limits, based on the interpretation of the derivative as the instantaneous rate of change. The principle behind this proof is found in a late nineteenth-century calculus textbook by J. M. Rice and W. W. Johnson, The Elements of the Differential Calculus, Founded on the Method of Rates or Fluxions (Wiley, New York, 1874). A spider walks with speed 1 in a circular path around the outside of a round satellite of radius 1, as shown in Figure 1. At time t the spider will have travelled a distance t , which corresponds to a central angle of t radians. The altitude of the spider, in the standard coordinate system, is and the spider is units to the right (or left) of the origin.

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