已知函数f(1+cotx)sinx^2-2sin(x+π/4)sin(x-π/4)

题目

已知函数f(1+cotx)sinx^2-2sin(x+π/4)sin(x-π/4)
1、若tana=2,求f(a)
2、若x∈[π/12,π/2],求f(x)的取值范围

答案

(1+cotx)sin^2x=sin^2x+sinxcosx
2sin(x+π/4)sin(x-π/4)=根号2(sinx+cosx)*根号2/2(sinx-cosx)=sin^2x-cos^2x
f(x)=(1+cotx)sinx^2-2sin(x+π/4)sin(x-π/4)=sinxcosx+cos^2x
f(x)=1/2sin2x+1/2(2cos^2x-1)+1/2
f(x)=1/2[sin2x+cos2x]+1/2
f(x)=根号2/2sin(2x+π/4)+1/2
1.tana=2=> sina/cosa=2 sin^2a+cos^2a=1=>cos^2a=1/5
f(a)=sinacosa+cos^2a=3cos^2a=3/5
2.x∈[π/12,π/2],2x属于[π/6,π]
2x+π/4属于[5/12π,5/4π]
sin(2x+π/4)属于[根号2/2,1]
f(x)取值范围为[1,(根号2+1)/2]