University Seminar- Computability, Complexity, and Algebraic Structure-Trigonometric Functions in the Second-Order Theory of Real Numbers: an Early Transcendentals Approach

Time: Wednesday, December 4, 2:20 – 3:20pm

Place: Rome Hall, Room 352

Speaker:  Keshav Srinivasan, GWU

Title: Trigonometric Functions in the Second-Order Theory of Real Numbers: an Early Transcendentals Approach

Abstract:  Conventional proofs that the derivative of sine is cosine are based on a squeeze theorem argument involving comparing the sides and area of a right triangle to the arc length and 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!  And even if this circularity is avoided, invoking arc length and area violates the spirit of an “early transcendentals” approach to calculus where derivatives of trigonometric functions are studied without relying on integrals.  We will discuss how both the concept of angle and trigonometric functions can be defined from first principles in the second-theory of Dedekind-complete ordered fields, taking inspiration from Euclid and Hilbert's axiomatizations of Euclidean geometry.  We will then present an argument adapted from the work of Gerson Robison in order to give a non-circular early-transcendentals proof of the derivative of sine.