已知函数f(x)=3sin²x+2√3sinxcosx+5cos²x.{1}求函数f【x】的周期和最大值.√=根号 {2}已知f【a】=5.求tana的值
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![已知函数f(x)=3sin²x+2√3sinxcosx+5cos²x.{1}求函数f【x】的周期和最大值.√=根号 {2}已知f【a】=5.求tana的值](/uploads/image/z/3020451-51-1.jpg?t=%E5%B7%B2%E7%9F%A5%E5%87%BD%E6%95%B0f%28x%29%3D3sin%26%23178%3Bx%2B2%E2%88%9A3sinxcosx%2B5cos%26%23178%3Bx.%7B1%7D%E6%B1%82%E5%87%BD%E6%95%B0f%E3%80%90x%E3%80%91%E7%9A%84%E5%91%A8%E6%9C%9F%E5%92%8C%E6%9C%80%E5%A4%A7%E5%80%BC.%E2%88%9A%3D%E6%A0%B9%E5%8F%B7+%7B2%7D%E5%B7%B2%E7%9F%A5f%E3%80%90a%E3%80%91%3D5.%E6%B1%82tana%E7%9A%84%E5%80%BC)
已知函数f(x)=3sin²x+2√3sinxcosx+5cos²x.{1}求函数f【x】的周期和最大值.√=根号 {2}已知f【a】=5.求tana的值
已知函数f(x)=3sin²x+2√3sinxcosx+5cos²x.{1}求函数f【x】的周期和最大值.
√=根号 {2}已知f【a】=5.求tana的值
已知函数f(x)=3sin²x+2√3sinxcosx+5cos²x.{1}求函数f【x】的周期和最大值.√=根号 {2}已知f【a】=5.求tana的值
做过的原题
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.
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.