I am facing a problem where I have two waves combined: \begin{equation} y = A\sin(b_1x)+B\cos(b_2x) \end{equation} Where $ b_1 $ and $ b_2 $ are non-rationals. i.e. \begin{align} & b_1 = \sqrt{3+\sqrt{5}} & b_2 = \sqrt{3-\sqrt{5}} \end{align}
I wanted to know if there exists $ X_p $ such that $ y(x+X_p) = y(x) $. I am confused here because if I actually try to solve this equation, the common procedure is to do the following: \begin{align} & y(x+X_p) = y(x) \\ & A\sin(b_1(x+X_p))+B\cos(b_2(x+X_p)) = A\sin(b_1x+2\pi n)+B\cos(b_2x+2\pi m ) \\ & \forall n,m \in \mathbb{N} \end{align} This will give 2 equations in $X_p$ , $n$ and $m$ if the sine and cosine are equated separately. However, I could not see a solution to that and I start to doubt whether a solution exists at all...
EDIT:
I am doubting a solution that can be applied to any A and B.
Thank you!