Converting Cartesian coordinates to polar coordinates

Conversion from Cartesian to Polar Coordinates

The Cartesian Coordinate System is a two-dimensional coordinate system in which the position of a point is specified by its distance along the x and y axis.
This grid system consists of two perpendicular number lines, usually labelled x (horizontal) and y (vertical). Each point on the plane corresponds to a pair of numbers, called coordinates.
The origin (0, 0) is the point of intersection of the x and y axis.

The Polar Coordinate System represents each point in the plane by a distance and an angle. The distance is the direct distance from the origin and the angle is the angle from the positive x-axis (or counterclockwise from the x-axis).
Polar coordinates are often represented as (r, θ), where r is the radial coordinate (distance from origin) and θ is the angular coordinate (angle with x-axis).

You can convert a point in the Cartesian coordinate system (x, y) to polar coordinates (r, θ) using the formulas: r = √(x² + y²) and θ = tan⁻¹(y/x).
If your Cartesian coordinates are positive (x > 0 and y > 0), you’re in the first quadrant, so your polar coordinates will be (r, θ).
If your x coordinate is negative (x < 0) and your y coordinate is positive (y > 0), you’re in the second quadrant, so your polar coordinates will be (r, θ + π).
If both your Cartesian coordinates are negative (x < 0 and y < 0), you’re in the third quadrant, so your polar coordinates will be (r, θ + π).
If your x coordinate is positive (x > 0) and your y coordinate is negative (y < 0), you’re in the fourth quadrant, so your polar coordinates will be (r, θ + 2π).

Polar coordinates make it easier to solve problems involving circular and spiral patterns or any situation where measurement from a central point is convenient.
They are particularly useful for understanding complex numbers, as they allow us to visualise them geometrically and manipulate them algebraically with more ease.