N 122.
1)
sin(x+60°)cos(x+30°) =1/2 ;
(1/2)*(sin(x+60°+x+30°) + sin(x+60°-x-30°) ) = 1/2 ;
sin(90°+2x) + sin30°= 1 ;
cos2x + 1/2 =1 ;
cos2x = 1/2 ;
2x = ±π/3 +2π*n , n∈ Z.
x = ±π/6 +π*n , n∈ Z.
ответ : ±π/6 +π*n , n∈ Z.
2)
sin3xsinx +cos4x =0 ; * * *cos4x=cos(3x+x) = cos3xcosx -sin3xsinx * * *
sin3xsinx +cos3xcosx -sin3xsinx =0 ;
cos3xcosx =0 ;
[ cos3x =0 ; cosx =0.⇔ [ 3x =π/2 +π*n ; x =π/2 +π*n , n∈ Z. ⇔
[ x = π/6 +(π/3)*n ; x =π/2 +π*n , n∈ Z. * * * π/2 +π*n ⊂ π/6 +(π/3)*n * * *
ответ : π/6 +(π/3)*n , n ∈ Z .
3)
cos6xcos3x =cos7xcos4x ;
(1/2) * ( cos(6x-3x) +cos(6x+3x) ) = (1/2) * ( cos(7x-4x) +cos(7x+4x) ) ;
cos3x +cos9x = cos3x +cos11x ;
cos11x -cos9x =0 ;
-2sin (11x -9x) /2 * sin(11x +9x) /2 =0
sinx * sin10x =0 ⇔ [ sinx =0 ; sin10x =0 . ⇒ [ x =πn ; 10x =πn , n n ∈ Z .
ответ : π*n /10 , n ∈ Z .
4)
12cos² (x/2) = 9 - 4cos(3x/2) * cos(x/2) ; * * * cos² (α/2) = (1+cosα) /2 * * *
12* (1+cosx) /2 = 9 - 4*(1/2)* ( cos(3x/2 -x/2) + cos(3x/2+x/2) ) ;
6 +6cosx =2cosx +2cos2x ;
cos2x - 2cosx -3 =0 ;
2cos²x -1 -2cosx -3 =0 ;
cos²x -cosx -2 =0 ; * * * замена : cosx = t ; -1 ≤ t ≤1 * * *
t² -t -2 =0 ;
t₁ = 2 > 1 не удовлетворяет
t ₂ = -1⇒ cosx = -1 ;
x = π +2πn , n ∈ Z .
ответ : π +2π*n , n∈ Z .