• Plot points using polar coordinates.
• Convert between polar coordinates and rectangular coordinates.
• Transform equations between polar and rectangular forms.
• Identify and graph polar equations by converting to rectangular equations.

Introduction to the Polar Coordinate Plane

In the rectangular coordinate system (or Cartesian plane), points are labeled [latex](x,y)[/latex] and plotted by moving horizontally and vertically from the origin. In this system, location is described in terms of horizontal and vertical distances.

In the polar coordinate system. Instead of moving along perpendicular axes, we locate points using a distance from the origin and an angle from a fixed direction.

polar coordinates

The polar coordinate plane includes:

1. The pole, corresponding to [latex](0,0)[/latex]
2. The polar axis
3. Distance, measured as [latex]r[/latex] and often marked by concentric circles centered at the pole.
4. Angles, marked using the unit circle angles.

In polar coordinates, each point is described by an ordered pair [latex](r,\theta)[/latex]

• [latex]r[/latex] is the radial distance from the pole.
• [latex]\theta[/latex] is the angle from the polar axis to the point’s location.

The same point can be represented in both systems. For example:

• In rectangular form: [latex](\sqrt(2),\sqrt(2))[/latex]
  $$ $$
• In polar form: [latex](2,\frac{pi}{4})[/latex]

Because angles can be measured in many ways (positive, negative, or adding multiples of $[latex]2\backslash pi[/latex]$), and distances can be negative (placing the point in the opposite direction), a single point often has many equivalent polar representations.