Suppose $-\frac{\pi}{2}+\epsilon\leq\theta\leq\frac{\pi}{2}-\epsilon$ for some $\epsilon>0$ and $x,y\in\mathbb{R}$. I am wondering whether there exists $\delta>0$ such that:
$$\left|\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)-\theta\right|\leq \left\{\begin{array}{ll}f(x,y)&~\text{if}~|x|,|y|\leq\delta\\c_2+f(x,y)&\text{otherwise}\end{array}\right.,$$
where $f(x,y)=|x\cos(\theta)-y\sin(\theta)+c(x^2+y^2)|$ and $c_1,c_2\geq0$ are constants.
Basically, I am wondering does there exist a small region around the origin where the error in approximating $|\theta|\leq\frac{\pi}{2}-\epsilon$ from the noisy in-phase and quadrature components $\sin(\theta)+x$ and $\cos(\theta)+y$ such that it 1) is governed by $f(x,y)$ if $(x,y)$ is in that region; and 2) is governed by a constant plus $f(x,y)$ everywhere else.
The two-dimensional Taylor series expansion of $\tan^{-1}\left(\frac{\sin(\theta)+x}{\cos(\theta)+y}\right)$ can be used to show that the first assertion is true, since $f(x,y)$ is essentially the first term (linear) and the remainder (quadratic) in this expansion, and it converges for $|x|,|y|$ small enough.
I am having trouble with the second assertion. It seems true by inspections of the plots, with $c_2\leq\pi$. Unfortunately, I have no idea how to prove this. Can anyone help?
This is related to my previous question.