Polar Coordinate Systems

Thus far we have used what is known as a Cartesian coordinate system, or a two-dimensional plane to represent curves with coordinate pairs $(x, y)$. Now we are going to learn about a new coordinate system known as the Polar Coordinate System, which uses coordinate pairs $(r, \theta)$, where $r$ represents the length of the line segment originating from the origin and terminating at the length of $r$, and where $\theta$ represents the angle the line segment makes with what we call the polar axis (analogous to the $x$-axis). For example, let's look at the following polar coordinates where $r = 2$, and $\theta = \frac{\pi}{3}$ or rather coordinates $P(2, \frac{\pi}{3} )$:

The horizontal black line represents our polar axis, while the vertical black line is there just to define the angles of $\frac{\pi}{2}$ and $\frac{3\pi}{2}$. Also notice that the line segment is dotted. The line segment itself is not a part of the coordinate $(2, \frac{\pi}{3} )$, but it is instead just an indicator of how far away from the origin a point $P$ should be.

The picture above illustrates the important of $r$, as it tells us the length away from the origin that a point $P$ is, while the angle $\theta$ describes the inclination or declination of that line segment. Now imagine a line segment of length $r$ sweeping across the origin. Because the line segment is of a constant length $r$, a full $360^{\circ}$ degree or $2\pi$ sweep around the origin will result in a circle. However, $\theta$ describes the position on the circle's perimeter to which the point lies on.

Now let's observe when both $r$ and $\theta$ increases:

As $r$ increases, the distance away from the origin increases, and as $\theta$ increases, the location of the point changes. You might be asking about values of theta such as $\theta = 3 \pi$, since $2 \pi$ defines a circle. Well, we'll simply go around the circle once ($2 \pi$) and then half around ($\pi$).

Hence, a polar coordinate $P$ can be represented in infinitely many ways. For example, the polar coordinates $P(3, \frac{\pi}{2})$ is the same as $Q(3, \frac{5\pi}{2} )$, etc…

So you might be asking now, can $r < 0$? The fact is, yes it can! If $r$ is negative, then imagine the line segment flipping around or going backwards from where it would have gone otherwise. For example:

$P(a, \frac{5\pi}{4})$ would position a point of length a away from the origin creating an angle of $\frac{5\pi}{4}$ with the polar axis. But $P(-a, \frac{5\pi}{4})$ positions the point of length $a$ in the opposite direction from the origin. Hence it follows that $P(a, k\pi) = P(-a, k\pi + \pi)$. For example:

Conversion Equations from (r, Θ) to (x, y)

Now suppose we have coordinates $(r, \theta)$ and want to plot them to cartesian coordinates. We need to first converse these coordinates from $(r, \theta)$ into $(x, y)$ using the following conversion equations:

(1)

\begin{align} x = r \cos \theta \quad \quad \quad y = r \sin \theta \end{align}

Recall the unit circle which say that $x = \cos \theta$ and $y = \sin \theta$. This is only true when $r = 1$, which is what the unit circle is defined by. Hence if $r$ increases, then $\cos \theta$ and $\sin \theta$ increase proportionally. That is $x = r \cos \theta$ and $y = r \sin \theta$. Hence $(x, y) = (r\cos \theta, r\sin \theta)$.

Conversion Equations From (x, y) to (r, Θ)

If we have cartesian coordinates $(x, y)$ and want to find their polar coordinates $(r, \theta)$, we can use the following equations:

(2)

\begin{align} r^2 = x^2 + y^2 \quad \quad \quad \tan \theta = \frac{y}{x} \end{align}