AI Tutor
Imagine standing in the center of an open field. To describe a tree to a friend, you wouldn't give grid coordinates like "walk 30 feet east, then 40 feet north" β€” you would point in a direction and say, "walk 50 feet that way."
That direct "direction and distance" approach is the foundation of polar coordinates.

The Polar Grid

In a polar coordinate system, we fix a central reference point called the pole (equivalent to the origin) and a horizontal ray pointing right called the polar axis (equivalent to the positive xx-axis).
Every point in the plane is named with an ordered pair (r,ΞΈ)(r, \theta), where rr is the directed distance from the pole and ΞΈ\theta is the directed angle measured counterclockwise from the polar axis.
So how do we actually place a point when given its coordinates?

Plotting Points with Positive Coordinates

Even though rr is written first in (r,ΞΈ)(r, \theta), it is usually easiest to locate the angle ΞΈ\theta first by rotating counterclockwise from the polar axis, and then count rr units along that terminal ray.
Let's plot the point A(4,5Ο€6)A\left(4, \frac{5\pi}{6}\right). First, find the angle ray 5Ο€6\frac{5\pi}{6} (150∘150^\circ in Quadrant II), then move out 44 circular rings along that ray.
What happens if our angle or our radius is negative?

Handling Negative Angles and Radii