在△ABC中,C=2AcosA=3/4,向量BA乘向量BC=27/2,则AC=

2acosa=bcosc+ccosb可由正弦定理得cosa=1/2,由余弦定理得bc=b方+c方-4,由重要不等式得bc小于等于4,再由重要不等式得b+c大于等于2倍根号下bc,所以b+c大于等于4

(1+tana)=2(1-tana) ,tana=1/3 ,cosa=3sina ,cos^2a=9sin^2a,cos^2a=9/10 ,sin^2a=1/10
sin2acosa-sina/cos2a=2sinacos^2a-sina/(1-2sin^2a)=(9/5-5/4)sina=11/20*√10/10
=11*√10/200

因为A为锐角,且tanA=1/2,算出tan(A/2)=√5－2 （√是根号）
(sin2AcosA-sinA)/(sin2A*cos2A)化简得到 1/2CosA
利用万能公式：
CosA={1-tan(2)x/2}/{1+tan(2)x/2} {tan(2)A就是tanA的平方}
带入就可以算出来了

(1) tan(π/4+a)=[tan(π/4)+tana]/[1-tan(π/4)tana]=2
1+tana=2(1-tana) tana=1/3
(2) (sin2acosa-sina)/cos2a
=sina[2(cosa)^2-1]/cos2a
=sinacos2a/cos2a
=sina=tana√[1+(tana)^2]
=√10/3

这是用极坐标表达的形式：ρ(θ)＝2acosθ;
直接利用极坐标面积公式：
╭π
S= (1/2)*∣ρ(θ)^2dθ
╯-π
╭π
= (1/2)*∣(2acosθ)^2dθ
╯-π
╭π
= a^2 *∣(1+cos2θ)dθ=(a^2)*2π
╯-π
即,结果为2(a^2)π

cos2acosa-(cos3a/2)^2
=cos2acosa-1/2(cos3a+1)
=cos2acosa-1/2(cos2acosa-sin2asina)-1/2
=1/2(cos2acosa+sin2asina)-1/2
=1/2cos(2a-a)-1/2
=(cosa-1)/2

(1) sin的平方2a+sin2acosa-cos2a=1
sin2acosa-cos2a=1-(sin2a)^2=(cos2a)^2
sin2acosa=cos2a(1+cos2a)
2sina(cosa)^2=cos2a*2(cosa)^2
sina=cos2a=1-2(sina)^2
2(sina)^2+sina-1
sina=-1,sina=1/2
∵a∈(0,π/2)
∴sina=1/2,a=π/6
tana=tan(π/6)=√3/3
(2)f(x)=sin的平方(x+a)+sin(a+x)cos(a+x)
=1/2[sin(2x+2a)-cos(2x+2a)+1]
=√2/2[sin(2x+2*π/6)cos(π/4)-cos(2x+2*π/6)sin(π/4)]+1/2
=√2/2sin(2x+2*π/6-π/4)+1/2
=√2/2sin(2x+π/12)+1/2
sin(2x+π/12)=1时,f(x)取得最大值
2x+π/12=2kπ+π/2
x=kπ+5π/24
因此,{x|x=kπ+5π/24}