Cos (2x-2 пи/3) + 5sin (x-пи/3) + 2=0

Cos (2x-2π/3) + 5sin (x-π/3) + 2=0

cos2 (x-π/3) + 5sin (x-π/3) + 2=0

cos² (x-π/3) - sin² (x-π/3) + 5sin (x-π/3) + 2=0

1-sin² (x-π/3) - sin² (x-π/3) + 5sin (x-π/3) + 2 = 0

1 - 2sin² (x-π/3) + 5sin (x-π/3) + 2=0

2sin² (x-π/3) - 5sin (x-π/3) - 3 = 0

sin (x-π/3) = y

2y² - 5y - 3=0

D=25 + 24=49

y₁=5 - 7 = - 1/2

4

y₂ = 5+7 = 3

4

При у = - 1/2

sin (x - π/3) = - 1/2

x-π/3 = (-1) ^ (n+1) * (π/6) + πn, n∈Z

x = (-1) ^ (n+1) * (π/6) + π/3 + πn, n∈Z

При у=3

sin (x-π/3) = 3

Так как 3∉[-1; 1], то

уравнение не имеет корней.

Ответ: (-1) ^ (n+1) * (π/6) + π/3 + πn, n∈Z.