Question

函数y=sinx-cosx+sinx*cosx的值域是

Answer 1

令t=tg(x/2),t为任意实数，
则由万能公式：sinα=2t/(1+t^2)，cosα=(1-t^2)/(1+t^2) ,
得：令f(t)=y
=sinx-cosx+sinx*cosx
=2t/(1+t^2)-(1-t^2)/(1+t^2)+[2t/(1+t^2)]*[(1-t^2)/(1+t^2)]
=1-[2(t+1)^2/(1+t^2)^2]
=1-2(t+1)^2/[(1+t)^2-2(1+t)+2]^2
=1-2/[(1+t)-2+2/(1+t)]^2
<1
再令g(t)=(1+t)+2/(1+t),则|g(t)|=|(1+t)+2/(1+t)|=|1+t|+|2/(1+t)|>=2根号[|1+t|*|2/(1+t)|]=2根号2，其中|1+t|=根号2时取等号
所以g(t)>=2根号2或者g(t)<=-2根号2
当t+1>=0时，f(t)=1-2/[(1+t)-2+2/(1+t)]^2=1-2/[g(t)-2]^2>=1-2/[2根号2-2]^2=-(1+2根号2)/2
当t+1<=0时，f(t)=1-2/[(1+t)-2+2/(1+t)]^2=1-2/[g(t)-2]^2>=1-2/[-2根号2-2]^2=(2根号2-1)/2
综上所述，-(1+2根号2)/2<=f(t)<1
所以y=sinx-cosx+sinx*cosx的值域是[-(1+2根号2)/2，1)