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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

fhguarni
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  • what is $w_1$ and $w_2$? – John Glenn Apr 05 '18 at 00:39
  • $\omega_{1}$ and $\omega_{2}$ are the angular frequency. $\omega_{1}=2 \pi f_{1}$. Frequency can be any value, that is why I did not write something like $\sin(71\pi t)$ or similar – fhguarni Apr 05 '18 at 01:03
  • You can use one of the trig identities on wikipedia, under linear combinations – John Glenn Apr 05 '18 at 03:23
  • Yes, I have already tried to do it. The most similar approach in my opinion is the rotation of vectors. But I get a even more complicated expression with t inside that I cannot solve. Any suggestion about an identity that I should use for this? – fhguarni Apr 05 '18 at 04:41

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