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]
and
[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.
Glossary
- 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
- cardioid
- 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]
- limaçon
- 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
- pole
- 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
- rose
- 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
