Polar Coordinates System

Polar Coordinates System

Polar coordinates are often more convenient than Cartesian coordinates when a problem involves circular symmetry or depends on distance from a fixed point and an angle from a reference direction.

Just as Cartesian coordinates use an ordered pair $(x, y)$, polar coordinates describe a point by a directed distance $r$ from a fixed origin $O$ (the pole) and an angle $\theta$ that the ray from the origin to the point makes with the polar axis (typically the positive $x$-axis). The location of a point $P$ is written $(r, \theta)$, where $r$ and $\theta$ are real numbers.

While $r$ is typically non-negative ($r \ge 0$), $r$ may also be negative.

• When $r > 0$, the angle $\theta$ measures any angle in standard position whose terminal side lies along the line from the origin to $P$. For example, $\bigl(5, \dfrac{\pi}{4}\bigr)$ is $5$ units from the pole at angle $\dfrac{\pi}{4}$.
• When $r < 0$, the point lies $|r|$ units in the opposite direction (at angle $\theta + \pi$). For example, $\bigl(-5, \dfrac{\pi}{4}\bigr)$ is $5$ units from the pole in the direction $\dfrac{\pi}{4} + \pi = \dfrac{5\pi}{4}$.

lightbulb

Note. $\bigl(-5, \dfrac{\pi}{4}\bigr)$ and $\bigl(5, \dfrac{5\pi}{4}\bigr)$ represent the same point.