The Spider's Spacewalk Derivation of sin' and cos'

Abstract

The 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 y = sm(t) and the spider is x = cos(t) units to the right (or left) of the origin. Now look how fast the spider is moving upward. Since the altitude of the spider at time t is y = sin(t), its upward velocity is y' = siri(t). Oops!?The spider loses its footing at time t, and since the gravity in outer space is negligible, it continues with the same direction and speed. It moves a distance 1 in additional time A? = 1,