题目

sinx+siny=根号2/2 cosx+cosy=根号6/2 则sin（x+y）=

提问时间：2020-11-03

答案

sinx+siny=√2/2,cosx+cosy=√6/2； 则sin(x+y)=
由sinx+siny=√2/2得sin²x+2sinxsiny+sin²y=1/2.(1)
由cosx+cosy=√6/2得cos²x+2cosxcosy+cos²y=3/2.(2)
(1)+(2)得2+2(cosxcosy+sinxsiny)=2+2cos(x-y)=2,故cos(x-y)=0；
(2)-(1)得cos²x-sin²x+2(cosxcosy-sinxsiny)+cos²y-sin²y=1
即有cos2x+2cos(x+y)+cos2y=2cos(x+y)+cos2x+cos2y
=2cos(x+y)+2cos(x+y)cos(x-y)=2cos(x+y)=1；
故cos(x+y)=1/2,x+y=π/3+2kπ.
∴sin(x+y)=sin(2kπ+π/3)=sin(π/3)=√3/2.