Polar coordinates
Some pictures are built around one center. Can a point be named by how far out it is and which way, instead of how far across and how far up?
Where you are 3 inside
What this area is about
Polar coordinates name a point (r, \theta): turn through the angle \theta from the polar axis, then walk the distance r. Here the pole sits at the origin, the polar axis along the positive x-axis, and angles are in radians unless a degree sign is written. r may be negative (walk |r| the opposite way), so every point has many polar names, and the pole is (0, \theta) for every \theta.
How these fit together
A procedure for single points. Plotting polar points sets up the pole, the polar axis and the polar grid, and shows why one point has many names.
Moving between systems. Converting between polar and rectangular moves one point at a time with x = r\cos\theta, y = r\sin\theta and x^2 + y^2 = r^2, in both directions. Converting an equation applies the same relations to a whole curve, so a circle like x^2 + y^2 = 49 becomes r = 7.
A path through it
Plot points first, then convert single points, then convert whole equations: each lesson uses the one before it.
This opens polar equations and their graphs, such as polar circles and cardioids, and the polar form of a complex number through modulus and argument. In calculus it leads to the slope of a polar curve.