**ACCURATE APPROXIMATIONS OF SOME EXPRESSIONS**

**INVOLVING TRIGONOMETRIC FUNCTIONS**

Marija Nenezi' c${}^{\,\mbox{\tiny 1)}}$, Branko Maleševi' c${}^{\mbox{\tiny 1)}}$, Cristinel Mortici${}^{\mbox{\tiny *},\mbox{\tiny 2)}}$

[*${}^{1)}$Faculty of Electrical Engineering, University of Belgrade,\
Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia\
${}^{2)}$Valahia University of Târgovişte, Bd. Unirii 18, 130082 Târgovişte;\
Academy of Romanian Scientists, Splaiul Independen ţei 54, 050094 Bucharest, Romania*]{}

[ The aim of this paper is to apply an original computation method due to Maleševi' c and Makragi' c [@[7]] to the problem of approximating some trigonometric functions. Inequalities of Wilker-Cusa-Huygens are discussed, but the method can be successfully applied to a wide class of problems. In particular, we improve the estimates recently obtained by Mortici [@[1]] and moreover we show that they hold true also on some extended intervals.]{}

[ Wilker inequality, trigonometric approximation]{}

Introduction
============

In the reference [@[2]] [J.$\,$B. Wilker]{} presented the inequality $$\label{Ineq_1}
2<\left(\nfrac{\sin x}{x}\right)^{\!2} + \nfrac{\tan x}{x},$$ for $x \!\in\! \left(0, \pi/2\right)$ and he asked for largest constant $c$ in $$\label{Ineq_2}
2+c x^3 \tan x
<
\left(\nfrac{\sin x}{x} \right)^{\!2} + \nfrac{\tan x}{x}\quad \mbox {for} \quad c>0,$$ and for $x \!\in\! \left(0, \pi/2\right)$. Recently, Wilker inequality is a lot studied in different paper works. In the paper [@[4]], [J.S.Sumner]{}, [A.A.Jagers]{}, [M. 