Polar Coordinate System

Calculus 2 · Axiom Academy

LESSON The Polar Coordinate System Locating a point not by how far across and up, but by a distance and a direction — and converting freely between the two. 1. A Point Is a Distance and a Direction Every point gets an ordered pair . Read it like a set of directions from the origin (the pole ): first turn to the angle , then walk r units out along that ray. The animation does exactly that for — it sweeps the angle, then extends the radius to land on the point. — angle from the polar axis, counterclockwise 2. The Bridge: One Right Triangle Drop a perpendicular from the point to the x -axis and a right triangle appears, with the radius r as its hypotenuse. The animation builds that triangle on and reads off its legs: the horizontal leg is , the vertical leg is . Run it backward — given the legs x and y — and the Pythagorean theorem returns r while returns . The ratio is the same for a point and its mirror through the origin, so alone can land you in the wrong half-plane. looks at the signs of x and y to choose the correct quadrant. Example: (1,1) gives and . 3. One Point, Infinitely Many Names In Cartesian coordinates each point has exactly one address. Polar coordinates do not — the animation lands the radius on , then reaches the identical spot two more ways: spin a full extra turn ( ), or point the radius the opposite direction and step backward with a negative r ( -r , ). All three arrows end on the same dot. Angles repeat every , so for any integer n .

This is the written version of the interactive lesson above. See the full Calculus 2 course.