Rectangular coordinates say "go across, then up." There is another natural way to give directions: "face this way, then walk this far." That is what you did with the unit circle, and it works for any distance from the origin, which in this setting is called the pole.
Idea clave
The polar coordinates $(r, \theta)$ name the point you reach by facing angle $\theta$ (from the positive $x$-axis) and moving $r$ units. If $r < 0$, move $|r|$ units backward, onto the opposite ray. Every point has many names: $$(r, \theta) = (r, \theta + 2k\pi) = (-r, \theta + \pi + 2k\pi).$$
Por qué funciona
Adding a full turn to $\theta$ points you along the same ray, so the point does not change. Walking backward along the ray at angle $\theta$ is the same as walking forward along the ray at angle $\theta + \pi$, which points the opposite way. So $(-r, \theta)$ and $(r, \theta + \pi)$ are the same point.
Worked examples
Ejemplo resuelto: Plotting
Plot $\left(3, \frac{5\pi}{6}\right)$.
- Face $\frac{5\pi}{6} = 150^\circ$, up and to the left.
- Move $3$ units along that ray. The point is in Quadrant II.
Respuesta: In Quadrant II, $3$ units from the pole
Ejemplo resuelto: Other names
Give three other polar names for $\left(2, \frac{\pi}{3}\right)$.
- Add a full turn: $\left(2, \frac{7\pi}{3}\right)$. Subtract one: $\left(2, -\frac{5\pi}{3}\right)$.
- Use a negative $r$ with the opposite angle: $\left(-2, \frac{\pi}{3} + \pi\right) = \left(-2, \frac{4\pi}{3}\right)$.
Respuesta: $\left(2, \frac{7\pi}{3}\right)$, $\left(2, -\frac{5\pi}{3}\right)$, $\left(-2, \frac{4\pi}{3}\right)$
Ejemplo resuelto: A negative $r$
Where is $\left(-4, \frac{\pi}{4}\right)$?
- Face $\frac{\pi}{4}$ (Quadrant I), then walk $4$ units backward.
- You land on the ray at $\frac{\pi}{4} + \pi = \frac{5\pi}{4}$, in Quadrant III.
Respuesta: Quadrant III; the same point as $\left(4, \frac{5\pi}{4}\right)$
Error común
Forgetting that one point has many polar names, and so marking $\left(-2, \frac{4\pi}{3}\right)$ wrong when it equals $\left(2, \frac{\pi}{3}\right)$. To compare two names, rewrite both with $r > 0$ and $0 \le \theta < 2\pi$. In rectangular coordinates each point has exactly one name; in polar coordinates it never does.