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

4) sin 2x = - sin x

x={0, 2pi/3, pi, 4pi/3}

Step-by-step explanation:

First, move sin x by adding sin x to both sides. sin 2x + sin x = 0. Next using the double angle identity, sin 2x=2 (sin x) (cos x), so 2 (sin x) (cos x) + sinx=0. Factoring, sin x (2cos x + 1) = 0. Solving, sin x=0 and 2cos x + 1 = 0, or cos x = - 1/2.

sin x = 0. Using the unit circle, sin x=0 when x=0 and x=pi.

cosx=-1/2. Using the unit circle, cos x = - 1/2 when x=2pi/3 and x=4pi/3.