To solve the equation cot(x)/3 = -√2/3, we can start by multiplying both sides of the equation by 3 to eliminate the fraction:

cot(x) = -√2

Next, we can take the reciprocal of both sides to find the value of x:

tan(x) = -1/√2

Since tan(x) is equal to -1/√2, we can find the reference angle by taking the inverse tangent of -1/√2:

tan^(-1)(-1/√2) ≈ -π/4

Since the tangent function is negative in the second and fourth quadrants, we can add π to the reference angle to find the solutions in those quadrants:

x = -π/4 + π = 3π/4

Therefore, the solutions to the equation cot(x)/3 = -√2/3 are x = -π/4 + 2πn and x = 3π/4 + 2πn, where n is an integer.