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Table 1 We present the first orders of Pade approximant for sin x , cos x , tan x

From: Padé approximant related to remarkable inequalities involving trigonometric functions

Function

Padé approximant

Associated Taylor polynomials

sinx

\(\sin _{[1/2]}(x)=\frac{6x}{6+x^{2}}\)

\(x-\frac{x^{3}}{6}\)

sinx

\(\sin _{[5/2]}(x)=\frac{2{,}520x-360x^{3}+11x^{5}}{2{,}520+60x^{2}}\)

\(x-\frac{x^{3}}{6}+\frac{x^{5}}{120}-\frac{x^{7}}{5{,}040}\)

cosx

\(\cos _{[2/2]}(x)=\frac{12-5x^{2}}{12+x^{2}}\)

\(1-\frac{x^{2}}{2}+\frac{x^{4}}{24}\)

cosx

\(\cos _{[4/2]}(x)=\frac{120-56x^{2}+3x^{4}}{120+4x^{2}}\)

\(1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}\)

cosx

\(\cos _{[4/4]}(x)=\frac{1{,}080-480x^{2}+17x^{4}}{1{,}080+60x^{2}+2x^{4}} \)

\(1-\frac{x^{2}}{2}+\frac{x^{4}}{24}-\frac{x^{6}}{720}\)

tanx

\(\tan _{[3/2]}(x)=\frac{15x-x^{3}}{15-6x^{2}}\)

\(x+\frac{x^{3}}{3}+\frac{2x^{5}}{15}\)