Summary of Polar Coordinates

Essential Concepts

  • The polar coordinate system provides an alternative way to locate points in the plane.
  • Convert points between rectangular and polar coordinates using the formulas

    [latex]x=r\cos\theta \text{ and }y=r\sin\theta [/latex]
    [latex]r^2={x}^{2}+{y}^{2}\text{ and }\tan\theta =\frac{y}{x}[/latex].
  • To sketch a polar curve from a given polar function, make a table of values and take advantage of periodic properties.
  • Use the conversion formulas to convert equations between rectangular and polar coordinates.
  • Identify symmetry in polar curves, which can occur through the pole, the horizontal axis, or the vertical axis.


angular coordinate
[latex]\theta [/latex] the angle formed by a line segment connecting the origin to a point in the polar coordinate system with the positive radial (x) axis, measured counterclockwise
a plane curve traced by a point on the perimeter of a circle that is rolling around a fixed circle of the same radius; the equation of a cardioid is [latex]r=a\left(1+\sin\theta \right)[/latex] or [latex]r=a\left(1+\cos\theta \right)[/latex]
the graph of the equation [latex]r=a+b\sin\theta [/latex] or [latex]r=a+b\cos\theta [/latex]. If [latex]a=b[/latex] then the graph is a cardioid
polar axis
the horizontal axis in the polar coordinate system corresponding to [latex]r\ge 0[/latex]
polar coordinate system
a system for locating points in the plane. The coordinates are [latex]r[/latex], the radial coordinate, and [latex]\theta [/latex], the angular coordinate
polar equation
an equation or function relating the radial coordinate to the angular coordinate in the polar coordinate system
the central point of the polar coordinate system, equivalent to the origin of a Cartesian system
radial coordinate
[latex]r[/latex] the coordinate in the polar coordinate system that measures the distance from a point in the plane to the pole
graph of the polar equation [latex]r=a\cos{n}\theta [/latex] or [latex]r=a\sin{n}\theta [/latex] for a positive constant [latex]a[/latex] and an integer [latex]n \ge 2[/latex]
space-filling curve
a curve that completely occupies a two-dimensional subset of the real plane