To solve this equation, we can first simplify it by using the double angle formula for cosine:

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

Substitute cos(2x) with 1 - 2sin^2(x) in the equation:

3sin(x)/5 + 2 = 2(1 - 2sin^2(x))/5
3sin(x)/5 + 2 = 2/5 - 4sin^2(x)/5

Now, we can combine like terms and move all terms to one side of the equation:

3sin(x)/5 + 4sin^2(x)/5 + 2/5 - 2 = 0
3sin(x) + 4sin^2(x) + 2 - 10 = 0
4sin^2(x) + 3sin(x) - 8 = 0

This is now a quadratic equation in terms of sin(x). We can solve this quadratic equation using the quadratic formula:

sin(x) = (-b ± √(b^2 - 4ac)) / 2a
sin(x) = (-(3) ± √((3)^2 - 4(4)(-8))) / (2(4))
sin(x) = (-3 ± √(9 + 128)) / 8
sin(x) = (-3 ± √137) / 8

Therefore, the solutions for sin(x) in this equation are (-3 + √137) / 8 and (-3 - √137) / 8.