Solving $\sin(6x) + \sin(4x) = 0$

Question

I have a test in a week and this will probably be one of the tasks. So an explanation would also come in handy.

They really need to do something about the questions, though. I need to type 1000 things but all I want is a solution and an explanation. I really don't know how to put that in a 100-word essay...

Answer 1

$$\sin6x+\sin4x=2\sin5x\cos x$$

So, $\sin 5x=0$ or $\cos x=0$

Another way:

\begin{align*} \sin6x&=-\sin 4x\\ \sin6x&=\sin (\pi+4x)\\ 6x&=n\pi+(-1)^n(\pi+4x)\\ x&=\frac{[n+(-1)^n]\pi}{6-4(-1)^n} \end{align*}