Converting Between Rectangular and Polar Coordinates Sometimes we want to change from rectangular coordinates to polar coordinates, and from polar coordinates to rectangular coordinates.

The drawing below will help us derive some conversion formulas.

To change to rectangular coordinates, use the formulas … To change to polar coordinates, use the formulas …

 Example 1: Change to rectangular coordinates:

 Example 2: Change to rectangular coordinates:

 Example 3: Change to polar coordinates: Note: when we solve an equation by taking the square root of both sides, we usually write a ±, but in this case we always want a non-negative value for r.

 Example 3: Change to polar coordinates: Note: there were two possible values for θ, but we chose the positive value, since the rectangular point is in quadrant II.

 Example 3: Change to polar coordinates: The polar point is given by …

 Example 4: Change to polar coordinates:

 Example 4: Change to polar coordinates: Note: there were two possible values for θ, but we chose the negative value, since the rectangular point is in quadrant III.

 Example 4: Change to polar coordinates: The polar point is given by …