Trigonometry Requires Calculus, Not Vice Versa (UMAP)

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 first-year 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