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In this video I go over one of my favorite topics in all of mathematics, and that is Polar Coordinates. A coordinate system represents a point in a plane by a pair of numbers called coordinates. In the basic Cartesian coordinate system, the pair of numbers (x, y) represent the horizontal and vertical distances perpendicular to the x and y axes. On the polar coordinate system represents an ordered pair (r, θ), in which the distance r is the distance from the origin O, and θ is the angle between the point and the “polar axis”. The polar axis is usually horizontal and corresponds to the positive x-axis in Cartesian coordinates. The angle θ is usually in radians. In this video I go over some more properties of the polar coordinate system, as well as the common definitions, such as a positive angle is counter-clockwise, etc., which I will be referencing in later videos. The polar coordinate system is much better suited to circular shapes such as circles, and other curves, because it better aligns with the nature of the curves, as opposed to the box-like Cartesian coordinate system. This is a VERY important introduction video into the wonderful world of Polar Coordinates so make sure to watch this video, as I will be exploring some truly amazing curves in later videos, so stay tuned for more!

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Polar Coordinates

A coordinate system represents a point in the plane by an ordered pair of numbers called coordinates.

Usually we use Cartesian coordinates, which are directed distances from two perpendicular axes.

Here we describe a coordinate system introduced by Newton, called the polar coordinate system, which is more convenient for many purposes.

We choose a point in the plane that is called the pole (or origin) and is labeled O.

Then we draw a ray (half-line) starting at O called the polar axis.

This axis is usually drawn horizontally to the right and corresponds to the positive x-axis in Cartesian coordinates.

If P is any other point in the plane, let r be the distance from O to P and let θ be the angle (usually measured in radians) between the polar axis and the line OP.

Then the point P is represented by the ordered pair (r , θ) and r, θ are called polar coordinates of P.

We use the convention that angle is positive if measured in the counter-clockwise direction from the polar axis and negative in the clockwise direction.

If P = O, then r = 0 and we agree that (0 , θ) represents the pole for any value of θ.

We extend the meaning of polar coordinates (r , θ) to the case which r is negative by agreeing that, as in the figure above, the points (-r , θ) and (r , θ) lie on the same line through O and at the same distance |r| from O, but on opposite sides of O.

If r > 0, the point (r , θ) lies in the same quadrant as θ; if r < 0, it lies in the quadrant on the opposite side of the pole.

Notice that (-r , θ) represents the same point as (r , θ + π).