Polar functions express mathematical relationships where each point on a curve is determined by an angle θ and a distance r from the origin. Instead of using Cartesian coordinates (x, y), polar equations describe curves using r = f(θ), which makes them especially suited for representing circular and spiral symmetries. As θ varies, the distance r changes according to the rule defined by the function, tracing intricate shapes such as spirals, roses, and cardioids. This system is powerful because i