Question

50 баллов! 17.ctg^2(2t-п/3)=3 18.tg^2(3x+П/2)=1/3 19. 3cos^2x-5cosx=0 20. знак модуля sin3x знак модуля =1/2

Answer 1

17)  ctq²(2x -π/3) =3;
1+cos(4x -2π/3) =3(1-cos(4x- 2π/3) ;
cos(4x - 2π/3) = 1/2 ;
4x -2π/3 = π/3 +2π*k ;
4x = π +2π*k;
x₁ =π/4 +π/2*k  ,k∈ Z ;
4x -2π/3 = - π/3 +2π*k ;
4x  = π/3 +2π*k ;
x₂=π/12 +π/2*k  , k∈ Z .
18) tq²(3x+π/2) =1/3 ;
ctq²3x =1/3 ;
3(1+cos6x) =1-cos6x ;
cos6x = -1/2 ;
6x =  (+/-)(π -π/3) +2π*k ;
x = (+/-)π/9 + π*k/3  ,  k∈ Z .
19) 
3cos²x -5cosx =0;
3cosx(cosx -5/3) =0 ;
cosx=0 ;
x=π/2 +π*k , k∈ Z.
cosx =5/3 >0. не имеет решения.
20) |sin3x| =1/2;
a) sin3x = -1/2;
 3x₁  =(-1)^(k+1)*π/6 + π*k  , k∈ Z.
 x₁  =(-1)^(k+1)*π/18 + π/3*k , k∈ Z.
b) sin3x = 1/2;
 3x₂  =(-1)^k*π/6 + π*k  , k∈ Z.
x₂  =(-1)^k*π/18 + π/3*k  , k∈ Z.