Question

sinx+siny=√2/2,求cosx+cosy的取值范围

Answer 1

令a=cosx+cosy

则a^2=(cosx)^2+(cosy)^2+2cosx*cosy

(sinx+siny)^2=(sinx)^2+(siny)^2+2sinx*siny=1/2

两式相加得a^2+1/2=2+2*(cos(x-y))

又因为1>cos(x-y)>-1

所以0〈（1/2+a2）〈4

即-1/2〈a2〈7/2

又（a2+1/2）恒大于零

所以0〈a2〈7/2

所以cosx+cosy的取值范围为[（-√14）/2,（√14）/2]