三角函数难题

3.
（1）
f(x)=4cosx(sinxcosπ/6+cosxsinπ/6)-1
=2√3sinxcosx+2cos²x-1
=√3sin2x+cos2x
=2(sin2x*√3/2+cos2x*1/2)
=2(sin2xcosπ/6+cos2xsinπ/6)
=2sin(2x+π/6)
∴f(x)的最小正周期：T=2π/2=π ,
（2）∵-π/6≤x≤π/4 ∴-π/6≤2x+π/6≤2π/3
∴2x+ π/6=π/2时,f(x)取得最大值 2
2x+ π/6=-π/6时,f(x)取得 最小值-1
4.
(1)f(x)＝1/2(1＋cos2ωx)＋√3/2*sin2ωx
＝1/2＋√3/2*sin2ωx+1/2*cos2ωx
＝1/2＋sin(2ωx＋π/6)
∵ f(x)的最小正周期为π,
∴T=2π/2ω＝π,解得ω＝1
∴ f(x)＝sin（2x＋π/6)＋1/2
∴f(2π/3)＝sin(4π/3+π/6)+1/2
=sin(3π/2)+1/2=1/2
(2)
由2kπ－π/2≤2x＋π/6≤2kπ＋π/2(k∈Z),
得:kπ－π/3≤x≤kπ＋π/6,(k∈Z)
∴函数f(x)的单调增区间为
[kπ－π/3,kπ＋π/6](k∈Z)
由 2kπ＋π/2≤2x＋π/6≤2kπ＋3π/2(k∈Z),
得：kπ＋π/6≤x≤kπ＋2π/3(k∈Z)
函数f(x)的单调减区间为
[kπ＋π/6,kπ＋2π/3](k∈Z)
由2x＋π/6＝kπ＋π/2(k∈Z)
得x＝kπ/2＋π/6(k∈Z)
∴f(x)图象的对称轴方程为
x＝kπ/2＋π/6(k∈Z)