converting between cartesian and polar coordinates

The Cartesian coordinate system specifies a point by the projection of its position to a set of orthogonal axes. Visually,

The polar coordinate system specifies a point by its distance from the pole and deviation from the polar axis. Visually,

Suppose that the coordinates $(x, y)$ and $(r, θ)$ represent the same point $P$ .
By convention, let the Cartesian origin $O$ be the same as the polar pole $O$ , and let $L$ be in the same relative orientation as the positive $x$ -axis.
Then we can construct the right triangle

which can be analyzed with trigonometric functions and the Pythagorean theorem to produce the identities

\begin{array}{ccc} x^{2} + y^{2} = r^{2} & \cos (θ) = \frac{x}{r} \\ \tan (θ) = \frac{y}{x} & \sin (θ) = \frac{y}{r} \end{array}

which can be rewritten

\begin{array}{ccc} r = \sqrt{x^{2} + y^{2}} & x = r \cos (θ) \\ θ = \arctan (\frac{y}{x}) & y = r \sin (θ) \end{array}

such that if I am given coordinates $(x, y)$ , I can convert to $(r, θ)$ with the first column identities,
and if I am given coordinates $(r, θ)$ , I can convert to $(x, y)$ with the second column identities.