sin55°怎么求?

先求sin(54度)的精确解,然后做逼近.
sin(54度)=cos(36度),设alpha=36度,则3*alpha=180度-2*alpha,即
cos(3*alpha)=-cos(2*alpha),由三角函数公式,
cos(3*alpha)=4*cos(alpha)^3-3*cos(alpha),
cos(2*alpha)=2*cos(alpha)^2-1,
联立后,用x代替cos(alpha),得一元三次方程,
4x^3+2x^2-3x-1=0,可分解(x+1)(4x^2-2x-1)=0.
得x=0.25*(1+根号5),另外两个根不在[0,1]区间内,实际上分别是cos180度和cos108度.
所以,sin(54度)=cos(36度)=0.25*(1+根号5).
cos(54度)=0.25*根号(10-2根号5).
sin(55度)=sin(54度+1度)=sin(54度)cos(1度)+cos(54度)sin(1度)
=0.25*(1+根号5)*(1-2*sin(0.5度)^2)+0.25*根号(10-2根号5)*sin(1度)
0.25*(1+根号5)*(1-2*(0.5/180*Pi)^2)+0.25*根号(10-2根号5)*(1/180*Pi) (对于小x,sin(x)~x,Pi为圆周率)
=0.81915256198.
如上计算,利用了加减乘除,开根号,和圆周率这些计算资料.
计算器算得
sin(55度)=0.81915204428.
相对误差6.3*10^(-7)（等同于0.000063%）,我们的估计已经无比精确.