15. Polar Coordinates

Rectangular coordinates $(x,y)$ are one way to specify a point, $P$, in the plane, but they are not the only way. When we are studying circles, it is useful to use Polar Coordinates $(r,\theta)$ which identify the point, $P$, using

• the radius $r$ and
• the polar angle $\theta$.

Like rectangular coordinates $P=(x,y)$, the polar coordinates can be written as an ordered pair, $P=(r,\theta)$. So you need to be careful to say whether this ordered pair is rectangular or polar.

Most often, the radius measures the (positive) distance from the origin, $O$, to the point, $P$, and the polar angle measures an angle (usually measured in radians and usually with $0 \le \theta \lt 2\pi$) counterclockwise from the positive $x$-axis to the ray $\overrightarrow{OP}$.

However, frequently the angle $\theta$ is allowed to be bigger than $2\pi$ or is allowed to be negative, in which case it is measured clockwise from the positve $x$-axis. Consequently, $\theta$ is non-unique; we can always add or subtract an arbitrary multiple of $2\pi$ or $360^\circ$. Thus the point with polar coordinates $(3,{30^\circ})$ can also be written as $(3,{390^\circ})$, or $(3,{750^\circ})$, or $(3,{-330^\circ})$, or $(3,{-690^\circ})$ all of which are shown in the plot with blue angles positive and red angles negative.

Further, once in a while, we allow $r$ to be negative, for example in the context of solving equations or graphing. Then $r$ is the negative of the distance from $O$ to $P$. When $r$ is negative, the point $(r,\theta)$ is obtained by going a distance $|r|$ along the ray at the angle $\theta\pm\pi=\theta\pm180^\circ$. Think of this as going backwards along the ray at the angle $\theta$. Thus the point with polar coordinates $(-3,{30^\circ})$ is actually the point $(3,{210^\circ})$ or $(3,-150^\circ)$.

Identify the point with each of the following coordinates: