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In this video I go over another example on Polar Coordinates, and this times show how to convert Cartesian coordinates to their polar counterpart. The difference between doing this, as opposed to converting polar to Cartesian coordinates is that we must be careful in selecting the correct quadrant when deciding on the angle. This is because the tanθ and the r^2 = x^2 + y^2 equations give two values for θ for when θ is between the range from 0 to 2pi. Also, the nature of the circular polar coordinate system, and the fact that we can use negative angles and negative distances, means that we can in fact write any Cartesian point into infinite representations when in Polar Coordinate form. This is a great example on seeing the differences in the steps taken when converting Cartesian to polar coordinates when compared with my last video, in which I converted polar to Cartesian coordinates, so make sure to watch this video!

Polar Coordinates playlist

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Polar Coordinates: Example 3: Cartesian to Polar

Example

Represent the point with Cartesian coordinates (1 , -1) in terms of polar coordinates.

Solution:

Note: The equations tan θ and r² = x² + y² does not uniquely determine θ when x and y are given because, as θ increases through the interval 0 ≤ θ ≤ 2π, each value of tan θ occurs twice.

Therefore, in converting from Cartesian to polar coordinates, it's not good enough just to find r and θ that satisfy tan θ and r² = x² + y².

And as in this example, we must choose θ (and r) so that the point (r , θ) lies in the correct quadrant.