Converting Polar and Rectangular

A polar point (r, theta) names a location by how far out you go and which direction you turn. But that same point sits somewhere on the xy-grid too, and a right triangle connects the two pictures. Drop a vertical line from the point down to the x-axis. That triangle has hypotenuse r, and the angle theta is measured from the positive x-axis. The horizontal leg is x, the vertical leg is y. That is the whole idea: one triangle, two coordinate systems.

The unit circle already gives you this triangle at r = 1, where cos(theta) and sin(theta) are defined as the x- and y-coordinates of the point on the circle at angle theta. Scale that triangle by r and the legs scale too: x = r cos(theta), y = r sin(theta). Going the other direction, the Pythagorean theorem gives r = √(x² + y²), since r is the hypotenuse. For the angle, tan(theta) = y/x, so theta = arctan(y/x) looks tempting, but arctan only returns angles between -π/2 and π/2, the range for a right-facing triangle. If your point actually sits with x negative, the true angle lands on the other side of the circle, and you have to add π to the arctan output to land in the correct quadrant.

This conversion is what lets the rest of the unit move freely between the two descriptions of the same curve. A spiral or rose written in polar form can be checked point by point against a rectangular graph, and equations like x² + y² = r² are exactly this conversion in disguise.