Conversion between Polar and Cartesian coordinates

Understanding Polar and Cartesian Co-ordinates

Polar co-ordinates are represented by (r, θ), where r is the radial distance from the origin and θ is the angle, counter-clockwise from the positive x-axis.
Cartesian co-ordinates are represented by (x, y), where x and y are distances along the x-axis and the y-axis respectively.
These two systems allow us to define any point in a plane, but can offer different conveniences depending on the problem at hand.

Converting from polar to Cartesian involves utilising the trigonometric relationships of a right triangle.
Given polar co-ordinates (r, θ), the corresponding Cartesian co-ordinates are calculated as: x = r cos(θ) and y = r sin(θ).
This takes advantage of the fact that cos(θ) gives the x-component of the unit vector in the direction of the angle θ, and sin(θ) gives the y-component.

Converting from Cartesian to polar requires understanding the Pythagorean theorem and the definition of tangent.
Given Cartesian co-ordinates (x, y), the magnitude of radial distance r can be calculated as: r = √(x² + y²).
The angle θ is then found by calculating the inverse tangent, or arctan of the ratio of y to x: θ = tan⁻¹(y/x).
However, you must take into account the quadrant of the Cartesian point. The tan⁻¹(y/x) value only gives the correct angle for a point in the first quadrant. For other quadrants, add π (or 180°) as necessary.

When converting from Cartesian to Polar, the quadrant of the origin point is essential for the correct determination of θ.
Points in the second and third quadrants will have a value of θ that is 180° away from the expected value, since the tan⁻¹(y/x) function only returns results spanning -90° to 90°.
Make sure to adjust the value of θ based on the quadrant of the point to get the accurate polar coordinates.