knrt.net
当前位置:首页 >> sin10Cos30sin50sin70 >>

sin10Cos30sin50sin70

sin(10)sin(30)sin(50)sin(70)=2sin(10)cos(10)sin(30)sin(50)sin(70)/[2cos(10)]=sin(20)sin(30)sin(50)cos(20)/[2cos(10)]=sin(40)sin(30)cos(40)/[4cos(10)]=sin(80)sin(30)/[8cos(10)]=sin(30)/8=1/16 用到的 sin(a)=cos(90-a) , sin(a)cos(a)=sin(2a)/2

sin10sin30sin50sin70=(sin10sin30sin50sin70*cos10)/cos10=(sin20sin30sin50sin70)/2cos10=(sin20sin30sin50cos20)/2cos10=(sin40sin30sin50)/4cos10=(sin40sin30cos40)/4cos10=(sin80sin30)/8cos10=(sin80sin30)/8sin80=1/16

解: sin10*sin30*sin50*sin70 =cos10*sin10*sin30*sin50*sin70/cos10 =1/2*sin20*sin70*sin30*sin50/cos10 =1/2*sin20*cos20*sin30*sin50/cos10 =1/4sin40*sin50*sin30/cos10 =1/4sin40*cos40*sin30/cos10 =1/8sin80*sin30/cos10 =1/8cos10*sin30/cos10 =1/8sin30 =1/16

sin10sin30sin50sin70 =sin30(sin10sin50sin70) =sin30(sin(60-10)sin10sin(60+10)) =sin30*(1/4sin30) =1/16 三倍角公式

由sin(90-α)=cosα 和 2sinαcosα=sin2α 得:sin10sin30sin50sin70=cos80cos60cos40cos20=cos80*1/2*cos40(cos20sin20)/sin20=1/2*cos80cos40*1/2*sin40/sin20=1/4*cos80*(cos40sin40)/sin20=1/4*cos80*1/2*sin80/sin20=1/8*cos80sin80/sin20=1/8*1/2*sin160/sin20=1/16sin(180-20)/sin20=1/16sin20/sin20=1/16

sin10°<cos70°<sin50°<cos30° 已知函数y=sinx在 [ 0,90°]上单增,cos70°=sin20°,cos30°=sin60° 故sin10°<sin20°<sin50°<sin60° 即sin10°<cos70°<sin50°<cos30°

sin10sin30sin50sin70 =1/2Cos20Cos40Cos80 =1/2*sin20Cos20Cos40Cos80/sin20 =1/4sin40Cos40Cos80/sin20 =1/8sin80cos80/sin20 =1/16sin160/sin20=1/16

因为sin70=cos20,sin50=cos40;sin20=2sin10cos10;sin80=cos10;所以 sin10sin30sin50sin70=sin10sin30sin50cos20=cos10sin10sin30sin50cos20/cos10=sin20sin30sin50cos20/2cos10=sin40sin30sin50/4cos10=sin40sin30cos40/4cos10=sin80sin30/8cos10=sin30/8=1/16

cos30°=sin60°,cos70°=sin20°,∵sin10°∴sin10°

cos10sin10sin30sin50sin70/cos10=(1/2)sin20sin30sin50cos20/cos10=(1/4)sin40sin30cos40/cos10=(1/8)sin80sin30/cos10=1/16

网站首页 | 网站地图
All rights reserved Powered by www.knrt.net
copyright ©right 2010-2021。
内容来自网络,如有侵犯请联系客服。zhit325@qq.com