f(x)=[2sin(x+π/3)+sinx]cosx-根号3sin2x,化简

=[2 sinx cosπ/3 +2cosx sinπ/3+sinx ]cosx-√3sin^2x
=[sinx+√3cosx+sinx]cosx-√3sin^2 x
=2sinxcosx+√3(cos^2x-sin^2x)
=sin 2x +√3cos 2x
=2(sin2x cosπ/3+ cos 2x sin π/3)
=2sin(2x+π/3)

（1）因为 f(x)=(2sinx+3cosx)cosx-3sin2x=sin2x+3cos2x= 2sin(2x+π3)
所以函数f（x）的图象的对称轴由下式确定： 2x+π3=kπ+π2,k∈Z
从而 x=k2π+π12,k∈Z．由题可知当k=0时,a有最小值 π12；
（2）当 x0∈[0,512π]时, 2x0+π3∈[π3,76π],
从而 sin(2x0+π3)∈[-12,1],则f（x0）∈[-1,2]
由mf（x0）-2=0可知：m≥1或m≤-2．

f(x)=1/2cos2x-根号3/2sin2x+1/2(2sin^2x-1+1)
=-根号3/2sin2x+1/2
最大值为(1+根号3)/2
最小正周期为 2π/2=π

f(x)=[2(sinx*1/2+cosx*√3/2)+sinx]cosx-√3sin²x
=(2sinx+√3cosx)cosx-√3sin²x
=2sinxcosx+√3(cos²x-sin²x)
=sin2x-√3cos2x
=2sin(2x-π/3)
T=2π/2=π

f(x)=[2sin(x+π/3)+sinx]cosx-根号3sin^2x
和差角公式展开
f(x)=(sinx+根号3cosx+sinx)cosx-根号3sin2x
=2sinx*cosx+根号3cosx*cosx-根号3sin2x
降幂公式
f(x)=sin2x+根号3*（1+cos2x)/2-根号3sin2x
=（1-根号3）sin2x+根号3/2cos2x+根号3/2
辅助角公式 不改变三角函数的周期
所以 最小正周期=π

f(x)=[2sin(x+π/3)+sinx]cosx-根3sin^2x
=[sinx+(√3）cosx+sinx]cosx-(√3）（sinx)^2
=2sinxcosx+(√3）[(cosx)^2-(sinx)^2]
=sin2x+(√3）cos2x
=2sin(2x+π/3),(x∈R).
1)函数f(x)的最小正周期是π.
2）x0∈[0,5π/12],
∴2x0+π/3∈[π/3,7π/6],
∴f(x0)的最大值是2,
∴m的取值范围是（2,+∞）.

f(x)=[2sin(x+π/3)+sinx]cosx-√3sin^2x
=[sinx+√3cosx+sinx]cosx-√3sin^2x
=2sinxcosx+√3cos^2x-√3sin^2x
=sin2x+√3cos2x
=2sin(2x+π/3)
f(x)的最小正周期为π
根据题意,m只需大于f(x)在该区间的最小值就行
当x=5π/12时,f(x)取最小值1
所以m>1

f(x)=[2sin(x+π/3)+sinx]cosx-√3sin²x =[2sinxcosπ/3+2cos2xsinπ/3+sinx]cosx-√3sin²x =2sinxcosx+√3(cos²x-sin²x) =sin2x+√3cos2x =2(1/2*sin2x+√3/2*cos2x) =2sin(2x+...

f(x)=cos(2x+π/3)+sin²X
=1/2*cos2x-√3/2*sin2x+(1/2)(1-cos2x)
=1/2-√3/2*sin2x,
(1)f(x)的最大值=（1+√3）/2.
最小正周期=π.
（2）由f(c/2)=－1/4得1/2-√3/2sinC=-1/4,
∴sinC=√3/2,C为锐角,
∴cosC=1/2,
cosB=1/3,
∴sinB=2√2/3,
∴sinA=sin(B+C)=sinBcosC+cosBsinC
=(2√2+√3）/6.