sin²x+cos²x=1
sin(2x)=2sinx*cosx
sin⁴x+cos⁴x=sin²(2x)-1/2
(sin²x+cos²x)²-2sinx*cosx=sin²(2x)-1/2;
sin²(2x)-1-sin(2x)-1/2=0
sin(2x)=t; ItI≤1;
t²-t-3/2=0
D=1+6=7
t=(1±√7)/2
t=(1+√7)/2∈∅
t=(1-√7)/2
sin(2x)=(1-√7)/2
2x=(-1)ⁿarcsin((1-√7)/2)+πn; n∈Z;
x=0.5*(-1)ⁿarcsin((1-√7)/2)+πn/2; n∈Z;
6) cosx+3sinx=2
cosx=cos²(x/2)-sin²(x/2)
sin(x)=2sin(x/2)*cos(x/2)
cos²(x/2)-sin²(x/2)+6sin(x/2)*cos(x/2)=2*(cos²(x/2)+sin²(x/2))
-cos²(x/2)-3sin²(x/2)+6sin(x/2)*cos(x/2)=0
разделим обе части на -cos²(x/2)≠0
1+3tg²(x/2)-6tg(x/2)=0
tg(x/2)=t
3t²-6t+1=0
D=36-12=24
t=(6+√24)/6=(1+√6/3)
t=(6-√24)/6=((1-√6/3)
tg(x/2)=(1+√6/3)
x/2=arctg(1+√6/3)+πn; n∈Z
x=2arctg(1+√6/3)+2πn; n∈Z
tg(x/2)=(1-√6/3)
x/2=arctg(1-√6/3)+πк; к∈Z
x=2arctg(1-√6/3)+2πк;к∈Z
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Verified answer
sin²x+cos²x=1
sin(2x)=2sinx*cosx
sin⁴x+cos⁴x=sin²(2x)-1/2
(sin²x+cos²x)²-2sinx*cosx=sin²(2x)-1/2;
sin²(2x)-1-sin(2x)-1/2=0
sin(2x)=t; ItI≤1;
t²-t-3/2=0
D=1+6=7
t=(1±√7)/2
t=(1+√7)/2∈∅
t=(1-√7)/2
sin(2x)=(1-√7)/2
2x=(-1)ⁿarcsin((1-√7)/2)+πn; n∈Z;
x=0.5*(-1)ⁿarcsin((1-√7)/2)+πn/2; n∈Z;
6) cosx+3sinx=2
cosx=cos²(x/2)-sin²(x/2)
sin(x)=2sin(x/2)*cos(x/2)
cos²(x/2)-sin²(x/2)+6sin(x/2)*cos(x/2)=2*(cos²(x/2)+sin²(x/2))
-cos²(x/2)-3sin²(x/2)+6sin(x/2)*cos(x/2)=0
разделим обе части на -cos²(x/2)≠0
1+3tg²(x/2)-6tg(x/2)=0
tg(x/2)=t
3t²-6t+1=0
D=36-12=24
t=(6+√24)/6=(1+√6/3)
t=(6-√24)/6=((1-√6/3)
tg(x/2)=(1+√6/3)
x/2=arctg(1+√6/3)+πn; n∈Z
x=2arctg(1+√6/3)+2πn; n∈Z
tg(x/2)=(1-√6/3)
x/2=arctg(1-√6/3)+πк; к∈Z
x=2arctg(1-√6/3)+2πк;к∈Z
(sin²x+cos²x)²-2sinx*cosx=sin²(2x)-1/2;
sin²(2x)-1-sin(2x)-1/2=0
sin(2x)=t; ItI≤1;
t²-t-3/2=0
D=1+6=7
t=(1±√7)/2
t=(1+√7)/2∈∅
t=(1-√7)/2
sin(2x)=(1-√7)/2
2x=(-1)ⁿarcsin((1-√7)/2)+πn; n∈Z;
x=0.5*(-1)ⁿarcsin((1-√7)/2)+πn/2; n∈Z;
sin(2x)=2sinx*cosx