Question

Find all solutions to the equation in the interval [0, 2π)

sin 2x = -sin x

Answer 1

sin 2x = 2sin x cos x

Answer 2

from the prob, sin 2x=-sin x, rearranging the term....
sin 2x + sin x =0

Answer 3

now apply the identity...
2 sin x cos x + sin x =0
sin x(2cos x + 1) =0 , hence
sin x=0 and 2 cos x +1 =0
x=arc sin 0 and 2cosx=-1,
cos x= (-1/2)
x= arc cos(-1/2)
x=0 and x=2pi/3.., where pi=3.1416