sin的逼近

hbghlyj

$f(x)=\dfrac{\frac{4x}{\pi}\left(1-\frac{x}{\pi}\right)}{\frac{5}{4}-\frac{x}{\pi}\left(1-\frac{x}{\pi}\right)},$则$\begin{array}l\sin x<f(x)\text{, 当}0<x<\fracπ6\text{或}\frac{5π}6<x<π.\\
\sin x>f(x)\text{, 当}\fracπ6<x<\frac{5π}6.\end{array}$

战巡

回复 1# hbghlyj

此类近似多了去了

参见“帕德近似”
比如$\sin(x)$在$\frac{\pi}{2}$处的2、2帕德近似为
\[\sin(x)\approx\frac{1-\frac{5}{12}(x-\frac{\pi}{2})^2}{1+\frac{1}{12}(x-\frac{\pi}{2})^2}\]

你想要精度更高还能调高次数，比如4、4帕德近似为
\[\sin(x)\approx\frac{1-\frac{115}{252}(x-\frac{\pi}{2})^2+\frac{313}{15120}(x-\frac{\pi}{2})^4}{1+\frac{11}{252}(x-\frac{\pi}{2})^2+\frac{13}{15120}(x-\frac{\pi}{2})^4}\]