1)x<-1 U x>1 -4(x²-1)-3≥1/(x²-1) (-4(x²-1)²-3(x²-1)-1)/(x²-1)≥0 (4(x²-1)²+3(x²-1)+1)/(x²-1)≤0 x²-1=a (4a²+3a+1)/a≥0 4a²+3a+1>0 при любом а,т.к D<0⇒a<0 x²-1<0⇒-1<x<1 не удов усл нет решения 2)-1<x<1 4(x²-1)-3≥1/(x²-1) (4(x²-1)²-3(x²-1)-1)/(x²-1)≥0 x²-1=a (4a²-3a-1)/a≥0 4a²-3a-1=0 D=9+16=25 a1=(3-5)/8=-1/4 U a2=(3+5)/8=1 a=0 _ + _ + [-1/4](0)[1] -1/4≤a<0 U a≥1 {x²-1≥-1/4⇒x²-3/4≥0⇒x≤-√3/2 U x≥√3/2 {x²-1<0⇒-1<x<1 -1<x≤-√3/2 U √3/2≤x<1 x²-1≥1⇒x²-2≥0⇒x≤-√2 U x≥√2 ответ x∈(-1;-√3/2] U [√3/2;1)
2cos²x-1-5cosx-2=0;⇒cosx=y;-1≤y≤1;⇒
2y²-5y-3=0;
y₁,₂=(5⁺₋√(25+24))/4=(5⁺₋7)/4;
y₁=(5+7)/4=3;⇒y₁>1⇒нет решения;
y₂=(5-7)/4=-1/2;⇒
cosx=-1/2;⇒x=⁺₋2π/3+2kπ;k∈Z.
2)1-cos8x=sin4x;⇒
sin²4x+cos²4x-cos²4x+sin²4x=sin4x;⇒
2sin²4x-sin4x=0;⇒
sin4x(2sin4x-1)=0;⇒
sin4x=0⇒4x=nπ;k∈Z;⇒x=nπ/4;n∈Z;
2sin4x-1=0;⇒
sin4x=1/2;
4x=(-1)ⁿ·π/6+nπ;n∈Z.
3)sin²x+4sinxcosx+3cos²x=0;⇒cos²x≠0 делим на cos²x:
tg²x+4tgx+3=0;tgx=y;⇒
y²+4y+3=0;
y₁,₂=-2⁺₋√(4-3)=-2⁺₋1;
y₁=-1;⇒
tgx=-1;⇒x=-π/4+nπ;n∈Z;
y₂=-3;⇒x=arctg(-3)+nπ;n∈Z.
4)cos4x-sin4x=-1/2;⇒cos4x=sin4x-1/2;⇒
cos²4x=sin²4x-2/2·sin4x+1/4;⇒
1-sin²4x-sin²4x+sin4x-1/4=0⇒
-2sin²4x+sin4x+3/4=0;⇒
sin4x=y;-1≤y≤1;
2y²-y-3/4=0;
y₁,₂=(1⁺₋√(1+6))/4=(1⁺₋√7)/4;
y₁=(1+√7)/4=(1+2.646)/4=0.9115;
sin4x=0.9115;⇒4x=(-1)ⁿarcsin(0.9115)+2nπ;n∈Z;
x=(-1)ⁿ(arcsin(0.9115))/4+nπ/2;n∈Z;
y₂=(1-2.646)/4=-0.4115;
4x=(-1)ⁿarcsin(-0.4115)+2nπ;n∈Z
x=[(-1)ⁿarcsin(-0.4115)+2nπ]/4.
-4(x²-1)-3≥1/(x²-1)
(-4(x²-1)²-3(x²-1)-1)/(x²-1)≥0
(4(x²-1)²+3(x²-1)+1)/(x²-1)≤0
x²-1=a
(4a²+3a+1)/a≥0
4a²+3a+1>0 при любом а,т.к D<0⇒a<0
x²-1<0⇒-1<x<1 не удов усл
нет решения
2)-1<x<1
4(x²-1)-3≥1/(x²-1)
(4(x²-1)²-3(x²-1)-1)/(x²-1)≥0
x²-1=a
(4a²-3a-1)/a≥0
4a²-3a-1=0
D=9+16=25
a1=(3-5)/8=-1/4 U a2=(3+5)/8=1
a=0
_ + _ +
[-1/4](0)[1]
-1/4≤a<0 U a≥1
{x²-1≥-1/4⇒x²-3/4≥0⇒x≤-√3/2 U x≥√3/2
{x²-1<0⇒-1<x<1
-1<x≤-√3/2 U √3/2≤x<1
x²-1≥1⇒x²-2≥0⇒x≤-√2 U x≥√2
ответ x∈(-1;-√3/2] U [√3/2;1)