How do we find out angle from $x$ & $y$ coordinates?

For any given point $(x, y)$, the angle say $\theta$ of the line, passing through this point & the origin, with the positive x-direction is given as $$\color{blue}{\tan\theta=\frac{y}{x}}$$ While other values are given as
$$\color{blue}{\sin\theta=\frac{y}{\sqrt{x^2+y^2}}}$$ $$\color{blue}{\cos\theta=\frac{x}{\sqrt{x^2+y^2}}}$$

Does the picture below help you visualise this? By 'angle' we mean $\theta$ below in plane polar coordinates. For some point $(x_0,y_0)$ on the plane, we can solve for $\theta$ using trigonometry.

In this recent answer, it is shown that $$ \theta=2\arctan\left(\vcenter{\frac y{x+\sqrt{x^2+y^2}}}\right) $$ This formula works for all $x,y$ except on the negative real axis, where $\theta$ goes from just under $\pi$ on top to just above $-\pi$ underneath.