1.求函数y=7-4sinxcosx+4cosx平方-4cosx四次方,的最大值和最小值2.已知函数f(x)=sin方wx+根号3sinwx(wx+π/2)的最小正周期为π.求函数f(x)在区间〔0,2π/3〕上的取值范围3.已知tan(π/4+α)=2,tanβ=1/2.(1)求tanα的
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1.求函数y=7-4sinxcosx+4cosx平方-4cosx四次方,的最大值和最小值2.已知函数f(x)=sin方wx+根号3sinwx(wx+π/2)的最小正周期为π.求函数f(x)在区间〔0,2π/3〕上的取值范围3.已知tan(π/4+α)=2,tanβ=1/2.(1)求tanα的
1.求函数y=7-4sinxcosx+4cosx平方-4cosx四次方,的最大值和最小值
2.已知函数f(x)=sin方wx+根号3sinwx(wx+π/2)的最小正周期为π.求函数f(x)在区间〔0,2π/3〕上的取值范围
3.已知tan(π/4+α)=2,tanβ=1/2.
(1)求tanα的值
(2)求sin(α+β)-2sinαcosβ/2sinαsinβ+cos(α+β)
1.求函数y=7-4sinxcosx+4cosx平方-4cosx四次方,的最大值和最小值2.已知函数f(x)=sin方wx+根号3sinwx(wx+π/2)的最小正周期为π.求函数f(x)在区间〔0,2π/3〕上的取值范围3.已知tan(π/4+α)=2,tanβ=1/2.(1)求tanα的
1)y=7-4sinxcosx+4cosx^2-4cosx^4
=7-4sinxcosx+4(1-cosx^2)cosx^2
=7-4sinxcosx+4sinx^2*cosx^2
=7-4sinxcosx+4(sinx*cosx)^2
=4(sinx*cosx)^2-4sinxcosx+1+6
=(2sinx*cosx-1)^2+6
=(sin2x-1)^2+6 因为(sin2x-1)^2∈[0,4],所以这个函数值域为[6,10]
2)这个估计你打的时候出了问题,第二个可能是sinwxsin(wx+π/2),而不是sinwx(wx+π/2),改后
f(x)=(1-cos2wx)/2+根号3 sinwx coswx=0.5-cos2wx/2+根3sin2wx/2
=cosπ/6sin2wx-sinπ/6cos2wx+0.5
=sin(2wx-π/6)+0.5,因为它的周期是π,所以w=1,
所以f(x)=sin(2x-π/6)+0.5
x∈〔0,2π/3〕 则 2x∈〔0,4π/3〕 则 2x-π/6∈〔-π/6,7π/6〕
则sin(2x-π/6)∈[-0.5,1] 则f(x)∈[0,1.5] 这个是第二题的答案
3)tan(π/4+α)=(tanπ/4+tanα)/(1-tanπ/4tanα)=(1+tanα)/(1-tanα)=2
所以 tanα+1=2-2tanα,解得tanα=1/3 这个是第一小题答案
sin(α+β)-2sinαcosβ/2sinαsinβ+cos(α+β)
=(sinαcosβ+cosαsinβ-2sinαcosβ)/(2sinαsinβ+cosαcosβ-sinαsinβ)
这个式子分子分母同除cosαcosβ得
原式=(tanα+tanβ-2tanα)/(2tanαtanβ+1-tanαtanβ)
=(tanβ-tanα)/(tanαtanβ+1)
因为tanα1/3,tanβ=1/2 所以原式=(1/6)/(7/6)=1/7
好累,纯粹手打,
1.
y=7-4sinxcosx+4cos^2x-4cos^4x
=7-2*(2sinxcosx)+4cos^2x(1-cos^2x)
=7-2sin2x+4cos^2x sin^2x
=7-2sin2x+(2sinxcosx)^2
=6+1-2sin2x+sin^2(2x)
=6+(1-sin2x)^2
-1 ≤ sin2x ≤ 1
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1.
y=7-4sinxcosx+4cos^2x-4cos^4x
=7-2*(2sinxcosx)+4cos^2x(1-cos^2x)
=7-2sin2x+4cos^2x sin^2x
=7-2sin2x+(2sinxcosx)^2
=6+1-2sin2x+sin^2(2x)
=6+(1-sin2x)^2
-1 ≤ sin2x ≤ 1
0 ≤ 1-sin2x ≤ 2
6 ≤ 6+(1-sin2x)^2 ≤ 10
最大值10,最小值6
2.
f(x)=sin^2(wx)+根号3sinwx sin(wx+π/2)
=1/2[1-cos(2wx)] + 根号3sin(wx) cos[π/2-(wx+π/2)]
=1/2-1/2 cos(2wx) + 根号3sin(wx) cos[-wx+]
=1/2-1/2 cos(2wx) + 根号3sin(wx) cos(wx)
=1/2-1/2 cos(2wx) + 根号3 /2 sin(2wx)
=根号3 /2 sin(2wx) -1/2 cos(2wx) +1/2
= sin(2wx)cosπ/6 -cos(2wx) sinπ/6 +1/2
= sin(2wx-π/6) +1/2
∵最小正周期为π
∴2π/(2w)=π,w=1
∴f(x) = sin(2x-π/6) +1/2
∵x∈〔0,2π/3〕
∴2x-π/6 ∈〔-π/6,7π/6〕
2x-π/6 = -π/6,或7π/6时,f(x)取最小值:-1/2+1/2=0
2x-π/6 = π/2时,f(x)取最大值:1+1/2=3/2
3.
tan(π/4+α)=2
(tanπ/4+tanα) / (1-tanπ/4 tanα) = 2
(1+tanα) / (1- tanα) = 2
1+tanα = 2- 2tanα
tanα=1/3
tanβ=1/2
[ sin(α+β)-2sinαcosβ ] / [ 2sinαsinβ+cos(α+β)]
=[ sinαcosβ+cosαsinβ-2sinαcosβ ] / [ 2sinαsinβ+cosαcosβ-sinαsinβ]
=[ cosαsinβ-sinαcosβ ] / [ sinαsinβ+cosαcosβ]
【分子分母同除以cosαcosβ:】
=(tanβ-tanα) / (tanαtanβ+1)
=(1/2-1/3)/(1/2*1/3+1)
=(1/6) / (1/7)
=1/7
收起
(1)y=7-4sinxcosx+4cosx^2-4cosx^4
=7-4sinxcosx+4(1-cosx^2)cosx^2
=7-4sinxcosx+4sinx^2*cosx^2
=7-4sinxcosx+4(sinx*cosx)^2
=4(sinx*cosx)^2-4sinxcosx+1+6
=(2sinx*cosx-1)^2+6
所以:6<=y<=10