已知f(x)=sin^2+2√3sinxcosx-cos^2 求f(x)单调增区间

来源:学生作业帮助网 编辑:六六作业网 时间:2024/12/23 16:06:05
已知f(x)=sin^2+2√3sinxcosx-cos^2求f(x)单调增区间已知f(x)=sin^2+2√3sinxcosx-cos^2求f(x)单调增区间已知f(x)=sin^2+2√3sinx

已知f(x)=sin^2+2√3sinxcosx-cos^2 求f(x)单调增区间
已知f(x)=sin^2+2√3sinxcosx-cos^2 求f(x)单调增区间

已知f(x)=sin^2+2√3sinxcosx-cos^2 求f(x)单调增区间
f(x)=sin²x+2√3sinxcosx-cos²x
=√3sin(2x)-cos(2x)
=2[(√3/2)sin(2x)-(1/2)cos(2x)]
=2sin(2x-π/6)
2kπ-π/2≤2x-π/6≤2kπ+π/2 (k∈Z)时,函数单调递增,此时kπ-π/6≤x≤kπ+π/3 (k∈Z)
函数的单调递增区间为[kπ-π/6,kπ+π/3] (k∈Z).

你好!
f(x)=根号3sin2x-cos2x
=2sin(2x-π/6)
当2x-π/6在(-π/2+2kπ,π/2+2kπ)时单调递增,解得x属于(-π/6+kπ,π/3+kπ)
当2x-π/6在(π/2+2kπ,3π/2+2kπ)时单调递减,解得x属于(π/3+2kπ,5π+2kπ) k属于Z