Polar Coordinates

Polar coordinates are simply a way to locate a point by saying which direction to face and how far to walk, instead of how far over and how far up. The name comes from the pole, the fixed center point (the origin) that every distance and angle is measured from. They're good for anything organized around a center: radar and sonar, microphone and antenna patterns, orbits, spirals. If you're standing at a lighthouse and want to say where a ship is, "that way, 3 miles out" beats "2.6 miles east and 1.5 north," and that's exactly $r$ (how far out from the origin) and $\theta$ (which way you're facing, measured counterclockwise from the positive x-axis). The four conversion formulas look like a lot to memorize, but they're just the unit circle and the Pythagorean theorem, which you already know.

The conversion formulas come straight from the unit circle. A point at distance $r$ and angle $\theta$ is just the unit-circle point $(\cos\theta, \sin\theta)$ stretched out by a factor of $r$, so $x = r\cos\theta$ and $y = r\sin\theta$. And $r^2 = x^2 + y^2$ is the Pythagorean theorem on the right triangle with legs $x$ and $y$.

One point, many names. In Cartesian coordinates every point has exactly one address. In polar, it doesn't. Spinning a full turn brings you back where you started, so $(2, \pi/6)$ and $(2, 13\pi/6)$ are the same point. Negative $r$ is allowed too, and it means "face angle $\theta$, then walk backward." So $(-2, 7\pi/6)$ is that same point again: face down and to the left, walk backward 2, and you land up and to the right.

Converting. To go from polar to Cartesian, use $x = r\cos\theta$, $y = r\sin\theta$. Going the other way, $r = \sqrt{x^2 + y^2}$ and $\tan\theta = y/x$.

A common mistake. It's tempting to get $\theta$ by typing $\arctan(y/x)$ into a calculator and stopping there. The trouble is that $\tan\theta$ alone can't tell $(1, 1)$ from $(-1, -1)$: both have $y/x = 1$, and arctan will always hand you an angle in the right half of the plane. Always check which quadrant the point is in, and add $\pi$ if you need to.

Why bother? Look at the table below. A circle centered at the origin is $x^2 + y^2 = a^2$ in Cartesian, but in polar it's just $r = a$: every point is the same distance out. Anything that's naturally organized around a center gets simpler. You'll see some of these shapes drawn further down.

Polar curves to know:

Radar and sonar report targets by distance and bearing, which are polar coordinates. Antenna radiation patterns, the orbits of planets, and the spiral arms of a sunflower are all naturally polar.