Some estimates of precision of the Huygens approximation

TL;DR

This paper analyzes the accuracy of the Huygens approximation for small angles by establishing bounds for the function using polynomial and rational functions.

Contribution

It provides new bounds and estimates for the precision of the Huygens approximation near zero, enhancing understanding of its accuracy.

Findings

Derived bounds for the Huygens function near zero

Compared polynomial and rational function approximations

Quantified the approximation's accuracy in a specific interval

Abstract

In this paper are given some estimates of precision of the Huygens approximation $x \approx \frac{2}{3} \sin x + \frac{1}{3} \tan x,$ for right neighbourhood of zero, by determining some boundaries for the Huygens function $f(x) = \frac{2}{3} \sin x + \frac{1}{3} \tan x,$ for $x \in \left(0, \frac{\pi}{2} \right)$, in forms of some polynomial and some rational functions.