6*cos(x) + 3*sin(2x) = 0,

Divide both sides by 3 and replace  sin(2x)  by  2*sin(x)*cos(x):  sin(2x) = 2*sin(x)*cos(x).  You will get

2*cos(x) + 2*sin(x)*cos(x) = 0.

Divide both sides by 2. You will get

cos(x) + sin(x)*cos(x) = 0.

Factorize: 

cos(x)*(1+sin(x)) = 0.

The equation deploys in two independent equations:


1)  cos(x) = 0  --->  x =   and/or  x = .


2)  1+sin(x) = 0  --->  sin(x) = -1  --->  x = .

Answer. The solutions are   and .