University Seminar- Computability, Complexity, and Algebraic Structure-The Derivative of Sine: an Early Transcendentals Proof in the Second-Order Theory of Real Numbers

Time: Wednesday, December 11, 2:20 – 3:20pm

Place: Rome Hall, Room 352

Speaker:  Keshav Srinivasan, GWU

Title: The Derivative of Sine: an Early Transcendentals Proof in the Second-Order Theory of Real Numbers

Abstract:  Conventional proofs that the derivative of sine is cosine are based on a squeeze theorem argument involving comparing the sides or areas of right triangles to the arc length or area of a circular arc, but the latter are defined in terms of integrals conventionally evaluated using trigonometric substitutions which presume that the derivative of sine is cosine!  There are some “late transcendentals” proofs that fix this issue, but we present a non-circular “early transcendentals” proof that does not depend on area, arc length, radians, pi, or other integral calculus-based concepts.  We will propose a first-principles definition of angles and trigonometric functions in the second-order theory of Dedekind-complete ordered fields inspired by Euclid and Hilbert's axiomatizations of geometry.  We will then present an argument adapted from the work of Gerson Robison in order to give a rigorous non-circular proof of the derivative of sine.