Question

△ABC中,向量AB*向量AC=8,记∠BAC=θ,△ABC的面积为S,且满足4*（2-√3）≤S≤4√3 （1）求θ的取值范围

（2）求函数f（θ）=2√3sin^2(π/4+θ)+2cos^2θ-√3的最大值和最小值

Answer 1

（1）向量AB*向量AC= │ AB│ │AC │cosθ
S=│ AB│ │AC │sinθ/2
∴S=4tanθ
∴4（2-√3）≤4tanθ≤4√3
2-√3≤tanθ≤√3
tan15°≤tanθ≤tan60°
又0°＜θ≤180°
∴15°≤θ≤60°即π/8≤θ≤π/3
(2)f（θ）=2√3sin^2(π/4+θ)+2cos^2θ-√3
=√3[1-cos(π/2+2θ)]+(cos2θ+1)-√3
=√3sin2θ+cos2θ+1
=2sin(2θ+π/6)+1
∵π/8≤θ≤π/3
∴5π/12≤2θ+π/6≤5π/6
∴2≤f≤3