Nested formulas for cosine and inverse cosine functions based on Vi\`ete's formula for $\pi$

TL;DR

This paper introduces nested formulas for cosine and inverse cosine functions based on Viète's formula for π, extending to complex arguments, hyperbolic functions, and other trigonometric functions, with numerical validation.

Contribution

It generalizes Viète's formula to nested representations of cosine and inverse cosine, including complex and hyperbolic cases, and develops algorithms for their computation.

Findings

Nested formulas valid for complex arguments and multiple branches

Numerical algorithms effectively compute these functions

Extensions to hyperbolic and other trigonometric functions demonstrated

Abstract

In this article, we develop nested representations for cosine and inverse cosine functions, which is a generalization of Vi\`{e}te's formula for $\pi$. We explore a natural inverse relationship between these representations and develop numerical algorithms to compute them. Throughout this article, we perform numerical computation for various test cases, and demonstrate that these nested formulas are valid for complex arguments and a $k$th branch. We further extend the presented results to hyperbolic cosine and logarithm functions, and using additional trigonometric identities, we explore the sine and tangent functions and their inverses.