已知函数f(x)=(sinx+cos)^2-2sin^2x求单调递减区间

来源:学生作业帮助网 编辑:作业帮 时间:2024/11/05 16:41:25

已知函数f(x)=(sinx+cos)^2-2sin^2x求单调递减区间
已知函数f(x)=(sinx+cos)^2-2sin^2x求单调递减区间

已知函数f(x)=(sinx+cos)^2-2sin^2x求单调递减区间
f(x)=(sinx+cosx)^2-2sin^2x
=sinx^2+cos^2+2sinxcosx-2sin^2x
=cos^2x-sin^2x+sin2x
=cos2x+sin2x
=√2sin(π/4+2x)
单调递减区间
2Kπ+π/2≤π/4+2x≤(2K+1)π+π/2
2Kπ+π/4≤2x≤2Kπ+5π/4
Kπ+π/8≤x≤Kπ+5π/8

f(x)=(sinx+cosx)^2-2sin^2x
=1+2sinxcosx-2(1-cos2x)/2
=2sin2x+2cos2x
=2√2sin(2x+π/4)
当2kπ+ π/2≤2x+π/4≤2kπ+3π/2,即kπ+ π/8≤x≤kπ+5π/8 时,
f(x)单调递减
=>f(x)的单调递减区间为:【kπ+ π/8,kπ+5π/8】