How to solve this equation $\sin 5x = \sin (x + \frac{\pi}{3})$

Question

Could you give a hint how to solve this equation $\sin 5x=\sin (x + \frac{\pi}{3})$?

I tried to change $\sin 5x$ in function of $\sin x$ and $\cos x$, but I wasn't able to go further.

Answer 1

hint If $$\sin(X)=\sin(a)$$ then

$$X=a+2k\pi$$

OR

$$X=\pi-a+2k\pi$$

Answer 2

you want

$$5x= 2k\pi + (x + \pi/3)$$ or

$$ 5x= 2k\pi + \pi -(x + \pi/3)$$

Therefore we have $$ x=(6k+1)\pi/{12} $$ or

$$ x= (3k+1)\pi/9$$ where $k$ is an arbitrary integer.

Answer 3

With Salahamam_Fatima's hint, suppose $a = x + \frac{\pi}{3}$

$sin(5x) = sin(\pi-(x+\frac{\pi}{3})+2k\pi)$

and

$sin(5x) = sin(x + \frac{\pi}{3}+2k\pi)$

Solve for x, and we get

$x = \frac{\pi}{9} + \frac{k\pi}{3}$ and $\frac{\pi}{12} + \frac{k\pi}{2}$

where k is an integer.