Polar and Cartesian coordinates

The distance between two points given in polar coordinates is found with the law of cosines.

Graphing the radius against the angle gives curves such as circles and roses.

A polar curve r = f(θ) is a parametric curve x = f(θ)cos θ, y = f(θ)sin θ with θ as the parameter, so polar points are positions given by a parameter.

The polar form r(cos θ + i sin θ) of a complex number uses the same distance r and angle θ as the polar coordinates of the point (a, b).

Conics have polar equations with a focus at the origin, so the same curve has Cartesian and polar forms.

Adding an angle to theta rotates a point about the origin, which a rotation matrix does to the coordinates.

Symmetry of sine and cosine shows in polar curves, such as symmetry across the axes.

The magnitude and direction of a vector are its polar coordinates r and theta.

A circle centered at the origin has polar equation r = constant.