Question

1) 3 tg(pi/4-x)<=корень 3 2) {sinx>=0 {cosx<=-корень 2/2 3) {cosx*cosy=1/4 {ctgx*ctgy=-3/4 4) cos2x-cos6x=0

Answer 1

1)tg(π/4-x)≤√3/3
π/6+πn≤π/4-x<π/2+πn
π/6-π/4+πn≤-x<π/2-π/4+πn
-π/24+πn≤-x<π/4+πn
-π/4+πn<x≤π/24+πn
x∈(-π/4+πn;π/24+πn]
2)2πn≤x≤π+2πn
3π/4+2πn≤x≤5π/4+2πn
x∈[3π/4+2πn;π+2πn}
3)cosxcosy=1/4⇒cos(x-y)+cos(x+y)=1/2
ctgxctgy--3/4⇒1/4:sinxsiny=-3/4⇒sinxsiny=-1/3⇒cos(x-y)-cos(x+y)=-2/3
прибавим и отнимем
2сos(x-y)=-1/6⇒cos(x-y)=-1/12⇒x-y=π-argcos1/12
2cos(x+y)=7/6⇒cos(x+y)=7/12⇒x+y=arccos7/12
прибавим и отнимем
2x=π-arccos1/12+arccos7/12⇒x=π/2-1/2arccos1/12+1/2arccos7/16
2y=π-arccos1/12-arccos7/12⇒x=π/2-1/2arccos1/12-1/2arccos7/16
4)2sin2xsin4x=0
sin2x=0⇒2x=πn⇒x=πn/2
sin4x=0⇒4x=πn⇒x=πn/4
Ответ x=πn/4