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Trigonometry Requires Calculus, Not Vice Versa (UMAP)
Yves Nievergelt

Mathematics Topic: Calculus 
Application Areas: All applications of measures of angles and trigonometry 
Prerequisites: About twothirds of a semester or else a onequarter course of calculus, including definitions of the concepts of limit, derivative, and integral. The contents of this Module can be sprinkled over lectures or assignments during the following terms of calculus. 

 ©1999 by COMAP, Inc.  Tools for Teaching 1998  37 pages 

After exposing the lack of foundation for trigonometry in the curriculum, this module explains one way to define and compute all the inverse trigonometric functions, inverse hyperbolic functions, and inverse exponenetial functions (logarithms). The theory and algorithms presented here involve only material at the level of firstyear calculus. Yet the topics developed might fit in courses from intermediate calculus (after limits, derivatives, and integrals, but possibly in parallel with transcendental functions) to advanced calculus.
Table of Contents:
INTRODUCTION
TRIGONOMETRY DEFINED BY CALCULUS 1
The Trigonometric Functions Arcsin and sin
The Trigonometric Functions Arccos and cos
Trigonometric Identities
TRIGONOMETRY DEFINED BY CALCULUS 2
The Trigonometric Functions Arctan and tan
Further Trigonometric Identities
ARCHIMEDES' ALGORITHM TO COMPUTE PI
BORCHARDT'S ALGORITHM
Borchardt's Algorithm for Inverse Trigonometric Functions
Borchardt's Algorithm for Inverse Hyperbolic Functions
Borchardt's Algorithm for the Natural Logarithm Function
CONCLUSIONS
SOLUTIONS TO SELECETED EXERCISES
ACKNOWLEDGMENT
REFERENCES
ABOUT THE AUTHOR



