Introduction to Polar Coordinates

AP Calculus · Axiom Academy

LESSON Introduction to Polar Coordinates A different way to locate points in the plane Step 1: Beyond Cartesian Coordinates You're used to identifying points with (x, y) coordinates. But there's another system: imagine standing at the origin and pointing in a direction (angle θ) at a certain distance (radius r). The angle is typically measured counterclockwise from the positive x-axis. This is especially useful for circular or spiral patterns, where distance and angle are the natural coordinates! Step 2: Converting Between Coordinate Systems Every point can be described in both systems! Here are the conversion formulas: Use the right triangle formed by r, θ, x, and y Be careful with the quadrant when finding θ Let's try an example: Convert the polar point (2, π/3) to Cartesian: So the Cartesian coordinates are (1, √3) In Cartesian coordinates, we write y = f(x). In polar coordinates, we write r as a function of θ: As θ varies, r tells us how far from the origin to go. The collection of all points (r, θ) traces out a curve! This is simple: "For any angle θ, go 3 units from the origin." This traces out a circle of radius 3 centered at the origin! As θ increases, r increases. So we spiral outward at a constant rate. Beautiful!

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