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.

for this case we must apply distributive property to the following expression:

the distributive property states that:

then, when applying the property we have to multiply the -2 for each term within the parenthesis: