To solve this equation, we can use the double angle identity for cosine and the sum to product identities for sine and cosine.

First, we rewrite the equation as:

2cos(3x) = 3sin(x) + cos(x)

Using the double angle identity for cosine, we have:

2(2cos^2(x) - 1) = 3sin(x) + cos(x)

Expanding on the left side, we get:

4cos^2(x) - 2 = 3sin(x) + cos(x)

Next, using the sum to product identities, we write the sine term in terms of cosine:

4cos^2(x) - 2 = 3(2sin(x)cos(x)) + cos(x)

4cos^2(x) - 2 = 6sin(x)cos(x) + cos(x)

Now, we can rewrite the equation in terms of cosine only:

4cos^2(x) - 2 = 6sin(x)cos(x) + cos(x)

4cos^2(x) - 2 = 6cos^2(x)sin(x) + cos(x)

Simplifying further:

4cos^2(x) - 2 = 6cos(x)sin(x) + cos(x)

4cos^2(x) - 2 = (6cos(x) + 1)sin(x)

Dividing both sides by (6cos(x) + 1):

(4cos^2(x) - 2) / (6cos(x) + 1) = sin(x)

Now, we have an expression for sin(x) in terms of cos(x). To solve for x, we can use the identity sin^2(x) + cos^2(x) = 1. By plugging in sin(x) from the above equation and solving for cos(x), we can find the values of x that satisfy the original equation.