Question

已知函数f(x)=3sin方x+2倍根号3sinxcosx+5cos方x.问函数f(x)的周期和最大值))

Answer 1

3sin方x=3（1-cos2x）/2
2倍根号3sinxcosx=根号3倍的sin2x
5cos方x=5（1+cos2x）/2
三项转化后相加整理可得f(x)=根号3倍的sin2x
+
cos2x
+4
=2sin(2x+π/6)+4
由此可得周期为π
最大值是6，当x取kπ+π/6

Answer 2

我刚答过的题:
(i)
f(x)=3sin^2x+2√3sinxcosx+5cos^2x
=sin^2
x+2√3sinxcosx+3cos^2
x+2(sin^2
x+cos^2
x)
=sin^2
x+2√3sinxcosx+(√3)^2·cos^2
x+2
=(sinx+√3cosx)^2+2
=[√(1^2+(√3)^2)·sin(x+arctan(√3/1)]^2+2
=[2sin(x+π/3)]^2+2
=4·sin^2
(x+π/3)
+2
∵sin(x+π/3)≤1
∴sin^2
(x+π/3)≤1
∴f(x)=4·sin^2
(x+π/3)
+2≤6;最大值是6.
f(x)=4·sin^2
(x+π/3)
+2
=2·[2sin^2
(x+π/3)-1]
+2
+2
=-2·[1-2sin^2
(x+π/3)]+4
=-2cos(2x+3π/3)+4
则f（x）的最小正周期是2π/2=π.
(ii)
代入原式:
f(a)=3sin^2
a+2√3sinacosa+5cos^2
a
=5;
∴3sin^2
a+2√3sinacosa=5(1-cos^2
a)=5sin^2
a;
2√3sinacosa=2sin^2
a;
√3sinacosa=sin^2
a;
两边同时除以sin^2
a得
√3/tana=1;
∴tana=√3.