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 | \(\tan _{[3/2]}(x)=\frac{15x-x^{3}}{15-6x^{2}}\) | \(x+\frac{x^{3}}{3}+\frac{2x^{5}}{15}\) |