# The Pentagonal Inequality

## TL;DR

This paper derives the exact maximum value of a linear combination of five or seven cosines with positive angles summing to pi, expressed as a rational function of the coefficients, using algebraic methods.

## Contribution

It provides a novel algebraic approach to determine the sharp bounds of cosine combinations with positive angles summing to pi.

## Key findings

- Derived explicit sharp bounds as positive real fractions.
- Extended the inequality analysis to five and seven cosine cases.
- Used algebraic transformations to simplify the bounds expression.

## Abstract

Given a positive linear combination of five (respectively seven) cosines, where the angles are positive and sum to pi, the aim of this article is to express the sharp bound of the combination as a Positive Real Fraction in the coefficients (hence cosine-free). The method uses algebraic and arithmetic manipulations with judicious transformations.