Find all solutions to $\sin(x+3)=\sin3x$

Question

Find all solutions to $\sin(x+3)=\sin3x$

I am trying to find real solutions. Do I need to solve $x+3=3x$?

Answer 1

HINT...you have $$x+3=3x+k.2\pi$$ or $$x+3=\pi-3x+k.2\pi$$

Answer 2

Option:

$\sin A -\sin B = $

$2(\sin(A-B)/2)(\cos(A+B)/2)= 0.$

Find the zeroes of $\sin(A-B)/2$ , and

of $ \cos(A+B)/2.$

Answer 3

Note that in general

$$\sin \alpha=\sin \beta \iff \alpha=\beta+2k\pi \quad \lor \quad \alpha=\pi-\beta+2k\pi$$

then

$$\sin(x+3)=\sin3x \iff x+3=3x+2k\pi \quad \lor \quad x+3=\pi-3x+2k\pi$$

and thus

$$\begin{cases}x+3=3x+2k\pi \implies x=\frac32+k\pi\\\\x+3=\pi-3x+2k\pi\implies x=\frac{\pi-3}4+k\frac{\pi}2\end{cases}$$