In Section AppCartesianPlane, we introduced the notion of assigning ordered pairs of real numbers called ‘coordinates’ to points in the plane. Recall the Cartesian coordinate plane is defined using two number lines – one horizontal and one vertical – which intersect at right angles at a point called the ‘origin’.

As seen below on the left, to plot a point with Cartesian coordinates, say \(P(-3,4)\), we start at the origin, travel horizontally to the left \(3\) units, then up \(4\) units. Alternatively, we could start at the origin, travel up \(4\) units, then to the left \(3\) units and arrive at the same location.

For the most part, the ‘motions’ of the Cartesian system (over and up) describe a rectangle, and most points can be thought of as the corner diagonally across the rectangle from the origin. For this reason, the Cartesian coordinates of a point are often called ‘rectangular’ coordinates.

In this section, we introduce a new system for assigning coordinates to points in the plane – polar coordinates as diagrammed above on the right. We start with an origin point, called the pole, and a ray called the polar axis.

We locate a point \(P\) using two coordinates, \((r,\theta )\), where \(r\) represents a directed distance from the pole and \(\theta \) is a measure of counter-clockwise rotation from the polar axis.

Roughly speaking, the polar coordinates \((r,\theta )\) of a point measure ‘how far out’ the point is from the pole (that’s \(r\)), and ‘how far to rotate’ from the polar axis, (that’s \(\theta \)).

For example, if we wished to plot the point \(P\) with polar coordinates \(\left (4, \frac {5\pi }{6}\right )\), we’d start at the pole, move out along the polar axis \(4\) units, then rotate \(\frac {5\pi }{6}\) radians counter-clockwise. The GeoGebra interactive below allows us to do just that by first adjusting the slider for the radius and then the slider for the angle.

We may also visualize this process by thinking of the rotation first. To plot \(P\left (4,\frac {5\pi }{6}\right )\) this way, we rotate \(\frac {5\pi }{6}\) counter-clockwise from the polar axis, then move outwards from the pole \(4\) units. Essentially we are locating a point on the terminal side of \(\frac {5\pi }{6}\) which is \(4\) units away from the pole. The sequence of diagrams below show this process.

If \(r < 0\), we begin by moving in the opposite direction on the polar axis from the pole. For example, to plot the point with polar coordinates \(Q\left (-3.5, \frac {\pi }{4}\right )\) we would start from the pole, move to the left \(3.5\) units, then rotate counter-clockwise \(\frac {\pi }{4}\) radians. Once again, we are fortunate to have a GeoGebra interactive to help us visualize this process.

If we interpret the angle first, we rotate \(\frac {\pi }{4}\) radians, then move back through the pole \(3.5\) units. Here we are locating a point \(3.5\) units away from the pole on the terminal side of \(\frac {5\pi }{4}\), not \(\frac {\pi }{4}\). This process is diagrammed below.

As you may have guessed, \(\theta < 0\) means the rotation away from the polar axis is clockwise instead of counter-clockwise. Hence, to plot \(R\left (3.5, -\frac {3\pi }{4}\right )\) we have the following.

From an ‘angles first’ approach, we rotate \(-\frac {3\pi }{4}\) then move out \(3.5\) units from the pole. We see that \(R\) is the point on the terminal side of \(\theta = -\frac {3\pi }{4}\) which is \(3.5\) units from the pole.

The points \(Q\) and \(R\) above are, in fact, the same point despite the fact that their polar coordinate representations are different. Unlike Cartesian coordinates where \((a,b)\) and \((c,d)\) represent the same point if and only if \(a=c\) and \(b=d\), a point can be represented by infinitely many polar coordinate pairs.

We explore this notion more in the following example.

In light of our work in Example plotpolarex, it should come as no surprise that any given point expressed in polar coordinates has infinitely many other representations in polar coordinates.

The following result characterizes when two sets of polar coordinates determine the same point in the plane. It could be considered as a definition or a theorem, depending on your point of view. We choose to state it as a property of the polar coordinate system.

Equivalent Representations of Points in Polar Coordinates

Suppose \(\left (r, \theta \right )\) and \(\left (r', \theta '\right )\) are polar coordinates where \(r \neq 0\), \(r' \neq 0\) and the angles are measured in radians. Then \(\left (r, \theta \right )\) and \(\left (r', \theta '\right )\) determine the same point \(P\) if and only if one of the following is true:

  • \(r' = r\) and \(\theta ' = \theta + 2\pi k\) for some integer \(k\)
  • \(r' = -r\) and \(\theta ' = \theta + (2k + 1) \pi \) for some integer \(k\)

All polar coordinates of the form \((0, \theta )\) represent the pole regardless of the value of \(\theta \).

The key to understanding this result, and indeed the whole polar coordinate system, is to keep in mind that \((r,\theta )\) means \((\text {directed distance from pole}, \text {angle of rotation})\).

If \(r = 0\), then no matter how much rotation is performed, the point never leaves the pole. Thus \((0, \theta )\) is the pole for all values of \(\theta \).

Now let’s assume that neither \(r\) nor \(r'\) is zero. If \(\left (r, \theta \right )\) and \(\left (r', \theta '\right )\) determine the same point \(P\) then the (non-zero) distance from \(P\) to the pole in each case must be the same. Since this distance is controlled by the first coordinate, we have that either \(r' = r\) or \(r' = -r\).

If \(r' = r\), then when plotting \(\left (r, \theta \right )\) and \(\left (r', \theta '\right )\), the angles \( \theta \) and \( \theta '\) have the same initial side. Hence, if \(\left (r, \theta \right )\) and \(\left (r', \theta '\right )\) determine the same point, we must have that \(\theta '\) is coterminal with \(\theta \). We know that this means \(\theta ' = \theta + 2\pi k\) for some integer \(k\), as required.

If, on the other hand, \(r' = -r\), then when plotting \(\left (r, \theta \right )\) and \(\left (r', \theta '\right )\), the initial side of \( \theta '\) is rotated \(\pi \) radians away from the initial side of \(\theta \). In this case, \( \theta '\) must be coterminal with \(\pi + \theta \). Hence, \( \theta ' = \pi + \theta + 2\pi k\) which we rewrite as \(\theta ' = \theta + (2k+1)\pi \) for some integer \(k\).

Conversely, if \(r' = r\) and \(\theta ' = \theta + 2\pi k\) for some integer \(k\), then the points \(P\left (r, \theta \right )\) and \(P'\left (r', \theta '\right )\) lie the same (directed) distance from the pole on the terminal sides of coterminal angles, and hence are the same point.

Now suppose \(r' = -r\) and \(\theta ' = \theta + (2k + 1) \pi \) for some integer \(k\). To plot \(P\), we first move a directed distance \(r\) from the pole; to plot \(P'\), our first step is to move the same distance from the pole as \(P\), but in the opposite direction. At this intermediate stage, we have two points equidistant from the pole rotated exactly \(\pi \) radians apart. Since \(\theta ' = \theta + (2k + 1) \pi = \left (\theta + \pi \right ) + 2\pi k \) for some integer \(k\), we see that \(\theta '\) is coterminal to \(\left (\theta + \pi \right )\) and it is this extra \(\pi \) radians of rotation which aligns the points \(P\) and \(P'\).

Next, we marry the polar coordinate system with the Cartesian (rectangular) coordinate system. To do so, we identify the pole and polar axis in the polar system to the origin and positive \(x\)-axis, respectively, in the rectangular system. We get the following result.

In the case \(r > 0\), Theorem polarrectangularconversion is an immediate consequence of Theorems cosinesinecircle and circularfunctionscircle.

If \(r < 0\), then we know an alternate representation for \((r,\theta )\) is \((-r, \theta + \pi )\). Since in this case, \(-r>0\), we know the theorem as stated is true for the representation \((-r, \theta + \pi )\) so we apply it here.

Applying Theorem polarrectangularconversion to \((-r,\theta +\pi )\) gives \(x = (-r) \cos (\theta + \pi ) = (-r)(-\cos (\theta )) = r\cos (\theta )\) as well as \(y = (-r) \sin (\theta + \pi ) = (-r)(-\sin (\theta )) = r\sin (\theta )\).

Moreover, \(x^2 + y^2 = (-r)^2 = r^2\), and \(\frac {y}{x} = \tan (\theta + \pi ) = \tan (\theta )\), so the theorem is true in this case, too.

The remaining case is \(r = 0\), in which case \((r,\theta ) = (0,\theta )\) is the pole. Since the pole is identified with the origin \((0,0)\) in rectangular coordinates, the theorem in this case amounts to checking ‘\(0=0\).’

Since we have argued that Theorem polarrectangularconversion is true in all cases, we put it to good use in the following example.

Now that we’ve had practice converting representations of points between the rectangular and polar coordinate systems, we now set about converting equations from one system to another.

Just as we’ve used equations in \(x\) and \(y\) to represent relations in rectangular coordinates (see Section Relations), equations in the variables \(r\) and \(\theta \) represent relations in polar coordinates. We convert equations between the two systems using Theorem polarrectangularconversion as the next example illustrates.

In practice, much of the pedantic verification of the equivalence of equations in Example eqnconversionex is left unsaid. Indeed, in most textbooks, squaring equations like \(r=-3\) to arrive at \(r^2=9\) happens without a second thought. Your instructor will ultimately decide how much, if any, justification is warranted.

If you take anything away from Example eqnconversionex, it should be that relatively simple equations in rectangular coordinates, such as \(y = x^2\), can become quite complicated in polar coordinates, and vice-versa.

In the next section, we devote our attention to graphing equations like the ones given in Example eqnconversionex number polareqntorect on the Cartesian coordinate plane without converting back to rectangular coordinates. If nothing else, number cardioidtorect above shows the price we pay if we insist on always converting to back to the more familiar rectangular coordinate system.