What Is the Solution to Sin(2x)+sin(x)=0?

The solutions to sin(2x)+sin(x)=0 is 0, 120 and 240 degrees. The sin(2x) can also be expressed as 2sin(x)cos(x), making the equation 2sin(x)cos(x)+sin(x)=0.

From there, sin can be factored out, leaving sin(x)(2cos(x)+1)=0. This means that either sin(x) or 2cos(x)+1 must equal zero. Sin(x) is only 0 at 0 degrees. To solve the other piece, 1 is subtracted from both sides, and both sides are divided by 2 to give cos(x)=-1/2. Cos(x) is -1/2 at 120 and 240 degrees. All three of these answers can be checked by inserting them back into the original equation. All three give an equation of 0=0, making them true.