How can I write $A\sin⁡(w_1t−k_1x) + B\sin(w_2t− k_2x)$ as$ 1$ trigonometric function?

Question

I am trying to do a problem and I need the frequency of the combined sine functions.

I am looking for the right trigonometric formula to do this.

Any suggestions would be appreciated.

Answer 1

HINT

Let use Arbitrary phase shift formula

$$a\sin x+b\sin(x+\theta)= c \sin(x+\varphi)$$

where

• $c = \sqrt{a^2 + b^2 + 2ab\cos \theta}$

and

• $\varphi = \operatorname{atan2} \left( b\,\sin\theta, a + b\cos\theta \right)$