Cartesian coordinates, the familiar x-and-y grid most people learn first, describe every point on a plane by how far across and how far up it sits relative to a fixed origin. Polar coordinates describe the exact same plane completely differently: every point is located instead by its straight-line distance from a central point and the angle of direction to reach it. Neither system is more fundamentally "correct" than the other — they're equally valid ways of pinpointing the same locations — but which one is more convenient depends enormously on the shape you're trying to describe.
A shape that hugs a centre is awkward in x and y
Consider a spiral, winding steadily outward from a central point. Describing that shape in Cartesian coordinates requires a genuinely awkward equation, since the relationship between horizontal and vertical position doesn't follow any simple, direct pattern as the spiral winds outward — the x and y coordinates both change in complicated, interrelated ways simultaneously. In polar coordinates, the same spiral often becomes remarkably simple to express: many spirals can be written as the distance from the centre increasing in direct, simple proportion to the angle swept out, a relationship that's awkward to express in x-y terms but almost trivially natural once distance-and-angle is the coordinate system doing the describing. Circles present an even starker case: a circle centred at the origin has an extremely simple polar equation (a constant distance, regardless of angle), while its Cartesian equation, though not difficult, is noticeably less direct.
A shape that hugs straight lines is awkward in polar
The relationship flips for shapes built from straight lines and right angles, like a square. A square's sides run at fixed, constant values of x or y, making its Cartesian equation almost trivial to write down directly. Expressed in polar coordinates, though, a square's boundary requires the distance from the centre to change in a comparatively complicated way depending on the angle, since a straight line generally doesn't correspond to any simple, constant relationship between polar distance and angle unless that line happens to pass directly through the origin. This complementary trade-off — spirals and circles favouring polar coordinates, straight-edged shapes favouring Cartesian ones — is exactly why mathematicians and engineers routinely switch between coordinate systems depending on which one makes the specific problem at hand mathematically simpler, rather than sticking rigidly to one system for every situation.
What we're still unsure about
The mathematics of converting between Cartesian and polar coordinates, and the fact that different coordinate systems suit different shapes, are completely settled mathematics, not in any dispute. What's more a matter of applied judgement than a fixed rule is recognising, in advance, which coordinate system will actually simplify a specific real problem, particularly in more advanced applications like physics and engineering, where a problem's natural symmetry (circular, spherical, or otherwise) often strongly suggests — but doesn't automatically guarantee — which coordinate system will make the resulting mathematics most tractable, a skill that generally develops through practice and pattern recognition rather than from a single formal rule that decides it every time.
This sits inside Polar Coordinates, one of seven topics in Trigonometry, one of seven domains in Mathematics, one of seventeen subjects the app can quiz you on.