若sinθ+cosθ=根号2 求log1/2(sinθ)·log1/2(cosθ)

问题描述:

若sinθ+cosθ=根号2 求log1/2(sinθ)·log1/2(cosθ)

sinθ+cosθ=根号2
两边平方,1+2sinθcosθ=2
2sinθcosθ=1
2sinθ(根号2-sinθ)=1
2sinθ^2-2根号2sinθ+1=0
sinθ=根号2/2
cosθ=根号(1-sinθ^2)=根号2/2
log1/2(sinθ)·log1/2(cosθ)
=(log1/2(根号2/2))^2
=(log1/2 (1/2)^(1/2))^2
=1/4