Objectives: In this tutorial, we define polar coordinates. We convert from polar coordinates to rectangular coordinates and from rectangular coordinates to polar coordinates. We graph some of the basic functions in polar coordinates using LiveMath and a graphing calculator. After working through these materials, the student should be able

• to convert from rectangular coordinates to polar coordinates;
• to convert from polar coordinates to rectangular coordinates;
• to recognize some standard polar graphs;
• to plot points in polar coordinates;
• to graph polar equations using a graphing calculator or software.

Modules:

 Definition. A point P in the plane has polar coordinates (r, q) if the line segment OP has length r and the angle that OP makes with the positive axis is q (measured in a counter clockwise direction). This definition requires that r > 0. If r < 0, then we consider the point Q which has polar coordinates (-r, q). Then the point P has polar coordinates (r, q) if P is the point on the straight line containing O and Q which is -r units from O on the opposite side of O from Q.

• Plotting points [using LiveMath].

Theorem. If a point P has polar coordinates (r, q) then the rectangular coordinates of P is (r cos(q), r sin(q)). In other words,

 x = r cos(q) y = r sin(q)

• Examples [using LiveMath].

Theorem. If a point P has rectangular coordinates (x, y) then the polar coordinates of P is (r, q) where

 r2 = x2 + y2 q = arctan(y/x).

We choose the positive square root of x2 + y2 for r if x > 0 and the negative square root otherwise.

• Examples [using LiveMath].

• Plotting Graphs

• Using the