Understanding Polar Coordinates: Learn It 2

Plotting Points in the Polar Plane

One key difference between polar and rectangular coordinates is that polar coordinates are not unique. A single point can be represented by infinitely many polar coordinate pairs.

Consider the point [latex]\left(1,\sqrt{3}\right)[/latex] in rectangular coordinates. This same point can be expressed in polar form as:

  • [latex]\left(2,\frac{\pi }{3}\right)[/latex]
  • [latex]\left(2,\frac{7\pi }{3}\right)[/latex]
  • [latex]\left(-2,\frac{4\pi }{3}\right)[/latex]

Let’s verify that [latex]\left(-2,\frac{4\pi }{3}\right)[/latex] represents [latex]\left(1,\sqrt{3}\right)[/latex]:

[latex]\begin{array}{ccccccc}\begin{array}{ccc}\hfill x& =\hfill & r\cos\theta \hfill \\ & =\hfill & -2\cos\left(\frac{4\pi }{3}\right)\hfill \\ & =\hfill & -2\left(-\frac{1}{2}\right)=1\hfill \end{array}\hfill & & & \text{and}\hfill & & & \begin{array}{ccc}\hfill y& =\hfill & r\sin\theta \hfill \\ & =\hfill & -2\sin\left(\frac{4\pi }{3}\right)\hfill \\ & =\hfill & -2\left(-\frac{\sqrt{3}}{2}\right)=\sqrt{3}.\hfill \end{array}\hfill \end{array}[/latex]

This confirms that negative values of [latex]r[/latex] are allowed in polar coordinates. While every point has infinitely many polar representations, each point has only one representation in rectangular coordinates.

The polar coordinate system has a clear visual interpretation. The value [latex]r[/latex] represents the directed distance from the origin to the point, while [latex]\theta[/latex] measures the angle that the line segment makes with the positive [latex]x[/latex]-axis.

In the polar plane, the horizontal line extending right from the center is called the polar axis (equivalent to the positive [latex]x[/latex]-axis). The center point is the pole or origin, corresponding to [latex]r=0[/latex]. Concentric circles represent points at fixed distances from the pole. The equation [latex]r=1[/latex] describes all points one unit from the pole, [latex]r=2[/latex] describes points two units away, and so on. Line segments radiating from the pole correspond to fixed angles.

A series of concentric circles is drawn with spokes indicating different values between 0 and 2π in increments of π/12. The first quadrant starts with 0 where the x-axis would be, then the next spoke is marked π/12, then π/6, π/4, π/3, 5π/12, π/2, and so on into the second, third, and fourth quadrants. The polar axis is noted near the former x-axis line.
Figure 2. The polar coordinate system.

How to: Plot Polar Points:

  1. Start with the angle [latex]\theta[/latex]. Measure counterclockwise from the polar axis if positive, clockwise if negative.
  2. If [latex]r > 0[/latex], move that distance along the terminal ray of the angle.
  3. If [latex]r < 0[/latex], move that distance along the ray opposite to the terminal ray.

Plot each of the following points on the polar plane.

  1. [latex]\left(2,\frac{\pi }{4}\right)[/latex]
  2. [latex]\left(-3,\frac{2\pi }{3}\right)[/latex]
  3. [latex]\left(4,\frac{5\pi }{4}\right)[/latex]

Watch the following video to see the worked solution to the example above.

For closed captioning, open the video on its original page by clicking the Youtube logo in the lower right-hand corner of the video display. In YouTube, the video will begin at the same starting point as this clip, but will continue playing until the very end.

You can view the transcript for this segmented clip of “7.3 Polar Coordinates” here (opens in new window).