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

Question

I found the following sentence.

To find the angle you use the arctangent function like this, angle $=\tan^{-1}\left(\frac{y}{x}\right)$.

But I am curious, is this the only way to know the angle? In other words, is it possible to find the angle with $\sin\left(\frac{y}{x}\right)$, $\cos\left(\frac{y}{x}\right)$ or $\tan$.. etc?

Answer 1

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
$$\color{blue}{\sin\theta=\frac{y}{\sqrt{x^2+y^2}}}$$ $$\color{blue}{\cos\theta=\frac{x}{\sqrt{x^2+y^2}}}$$

Answer 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.