Find solutions of the trigonometric equation sin 12x+cos 6x=0?

We'll change sin 12x, using the double angle identity,
into:

sin 12x = sin 2*(6x) = 2 sin 6x*cos
6x

We'll re-write the
equation:

2 sin 6x*cos 6x + cos 6x =
0

We'll factorize by cos
6x:

cos 6x(2 sin 6x + 1) =
0

We'll set each factor as
zero:

cos 6x = 0

6x =
+/-arccos 0 + 2kpi

6x = +/-(pi/2) +
2kpi

We'll divide by 6:

x =
+/-(pi/12) + kpi/3

2 sin 6x + 1 =
0

sin 6x = -1/2

6x =
(-1)^k*arcsin(-1/2) + kpi

x = (-1)^(k+1)*(pi/36) +
kpi/6

The solutions of trigonometric equation
are: {+/-(pi/12) + kpi/3 ; k integer}U{(-1)^(k+1)*(pi/36) + kpi/6 ; k
integer}.