trigonometry - Find t for $Asin(w_1t) = -Bsin(w_2t)$

I am trying to solve t for the next equation:

$$A\sin(\omega_{1}t)+B\sin(\omega_{2}t)=0$$

Where:

$A > B$

$\omega_{1}$ AND $\omega_{2}$ are constants

Reading in wikipedia about trigonometric identities and following this post: Identity for a weighted sum of sines / sines with different amplitudes

I tried:

$$A\sin(\omega_{1}t)+B\sin(\omega_{2}t)=C\sin(\omega_{1}t+\varkappa)=0$$

Where:

$$C^2=A^2+B^2+2AB\cos(\omega_{2}t-\omega_{1}t)$$

And

$$\varkappa=\arcsin\frac{B\sin(\omega_{2}t-\omega_{1}t)}{C}$$

if $C\sin(\omega_{1}t+\varkappa)=0$, means that $C$ and/or $sin(\omega_{1}t+\varkappa)$ are equal to $0$, so I can get a partial solution by making $C=0$:

$$t=\frac{\arccos-\frac{A^2+B^2}{2AB}}{\omega_{2}-\omega_{1}}$$

But for the expression $sin(\omega_{1}t+\varkappa)=0$ I have not been able to solve it.

Someone can help me with this please?

Thanks