1 + cos6x + 3sin3x = 0
1 + 1 - 2sin²3x + 3sin3x = 0
-2sin²3x + 3sin3x + 2 = 0
2sin²3x - 3sin3x - 2 = 0
2sin²3x - 4sin3x + sin3x - 2 = 0
2sin3x(sin3x - 2) + (sin3x - 2) = 0
(2sin3x - 1)(sin3x - 2) = 0
sin3x = -1/2; sin3x = 2 - не имеет решений
3x = (-1)ⁿ⁺¹π/6 + πn, n ∈ Z
x = (-1)ⁿ⁺¹π/18 + πn/3, n ∈ Z
Ответ: x = (-1)ⁿ⁺¹π/18 + πn/3, n ∈ Z.