Polar Coordinates

Locate points by radius and angle, and convert to and from rectangular form.

The idea

Polar coordinates locate a point by its distance from a fixed origin and the direction in which it lies — the natural description for a radar sweep, an orbit, or a ripple spreading from a point.

Fix an origin $O$, the pole, and a reference direction, taken to be the positive $x$-axis. To reach the point named $(r, \theta)$, rotate a ray from the reference direction through the angle $\theta$ and mark the point at distance $r$ along it.

Drop a perpendicular from that point to the $x$-axis. The result is a right triangle with hypotenuse $r$ and angle $\theta$ at the pole, and its two legs are the rectangular coordinates: $x = r\cos\theta, \qquad y = r\sin\theta.$

Unlike rectangular coordinates, polar names are not unique. Adding a full turn of $2\pi$ to $\theta$ returns the ray to the same position and names the same point, so every point except the pole has infinitely many names, and the pole itself is $r = 0$ at every angle. A negative $r$ is read as distance measured along the opposite ray.

Ways to work on it

• Walkthrough. Convert polar to rectangular and see why polar names repeat.
• Practice. Convert a polar point to a rectangular coordinate.
• Hardest. Convert rectangular to polar with the correct quadrant.

Not sure where to start? Take the ten-question placement test.