Solve the trigonometric equation sin 6x=-cos (3x), using double angle identity.

We'll use the double angle identity to re-write the
term sin 6x:

sin 6x = sin 2*(3x) = 2 sin 3x*cos
3x

We'll re-write the equation, moving all terms to one
side:

2 sin 3x*cos 3x + cos 3x =
0

We'll factorize by cos
3x:

cos 3x(2 sin 3x + 1) =
0

We'll set each factor as
zero:

cos 3x = 0

3x =
+/-arccos 0 + 2kpi

3x = +/-(pi/2) +
2kpi

We'll divide by 3:

x =
+/-(pi/6) + 2kpi/3

We'll set the next factor as
0:

2 sin 3x + 1 = 0

sin 3x =
-1/2

3x = (-1)^k*arcsin(-1/2) +
kpi

x = (-1)^(k+1)*(pi/18) +
kpi/3

The solutions of trigonometric equation
are: {+/-(pi/6) + 2kpi/3 ; k integer}U{(-1)^(k+1)*(pi/18) + kpi/3 ; k
integer}.