To solve the equation 2cos(x+π/3) = √3, we can start by isolating the cosine term.

Divide both sides of the equation by 2:
cos(x+π/3) = √3/2

Now, we need to find the values of x that satisfy this equation. We know that the cosine function has a value of √3/2 at π/6 and 11π/6. These are the reference angles for which the cosine is positive and equal to √3/2.

So, we can write two equations based on these reference angles:

x + π/3 = π/6 + 2πn (where n is an integer)
x + π/3 = 11π/6 + 2πn (where n is an integer)

Now, we can solve each equation for x:

For the first equation:
x = π/6 - π/3 + 2πn
x = -π/6 + 2πn

For the second equation:
x = 11π/6 - π/3 + 2πn
x = 5π/6 + 2πn

So, the general solutions for the equation 2cos(x+π/3) = √3 are:
x = -π/6 + 2πn, 5π/6 + 2πn (where n is an integer)