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Section9.4Introduction to Polar Coordinates

We are generally introduced to the idea of graphing curves by relating \(x\)-values to \(y\)-values through a function \(f\text{.}\) That is, we set \(y=f(x)\text{,}\) and plot lots of point pairs \((x,y)\) to get a good notion of how the curve looks. This method is useful but has limitations, not least of which is that curves that “fail the vertical line test” cannot be graphed without using multiple functions.

The previous two sections introduced and studied a new way of plotting points in the \(x,y\)-plane. Using parametric equations, \(x\) and \(y\) values are computed independently and then plotted together. This method allows us to graph an extraordinary range of curves. This section introduces yet another way to plot points in the plane: using polar coordinates.

Subsection9.4.1Polar Coordinates

Start with a point \(O\) in the plane called the pole (we will always identify this point with the origin). From the pole, draw a ray, called the initial ray (we will always draw this ray horizontally, identifying it with the positive \(x\)-axis). A point \(P\) in the plane is determined by the distance \(r\) that \(P\) is from \(O\text{,}\) and the angle \(\theta\) formed between the initial ray and the segment \(\overline{OP}\) (measured counter-clockwise). We record the distance and angle as an ordered pair \((r,\theta)\text{.}\) To avoid confusion with rectangular coordinates, we will denote polar coordinates with the letter \(P\text{,}\) as in \(P(r,\theta)\text{.}\) This is illustrated in Figure 9.4.1

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Figure9.4.1Illustrating polar coordinates.

Practice will make this process more clear.

Example9.4.2Plotting Polar Coordinates

Plot the following polar coordinates: \begin{equation*} A = P(1,\pi/4) B=P(1.5,\pi) C = P(2,-\pi/3) D = P(-1,\pi/4) \end{equation*}


Consider the following two points: \(A = P(1,\pi)\) and \(B = P(-1,0)\text{.}\) To locate \(A\text{,}\) go out 1 unit on the initial ray then rotate \(\pi\) radians; to locate \(B\text{,}\) go out \(-1\) units on the initial ray and don't rotate. One should see that \(A\) and \(B\) are located at the same point in the plane. We can also consider \(C=P(1,3\pi)\text{,}\) or \(D = P(1,-\pi)\text{;}\) all four of these points share the same location.

This ability to identify a point in the plane with multiple polar coordinates is both a “blessing” and a “curse.” We will see that it is beneficial as we can plot beautiful functions that intersect themselves (much like we saw with parametric functions). The unfortunate part of this is that it can be difficult to determine when this happens. We'll explore this more later in this section.

Subsection9.4.2Polar to Rectangular Conversion

It is useful to recognize both the rectangular (or, Cartesian) coordinates of a point in the plane and its polar coordinates. Figure 9.4.4 shows a point \(P\) in the plane with rectangular coordinates \((x,y)\) and polar coordinates \(P(r,\theta)\text{.}\) Using trigonometry, we can make the identities given in the following Key Idea.

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Figure9.4.4Converting between rectangular and polar coordinates.
Key Idea9.4.5Converting Between Rectangular and Polar Coordinates

Given the polar point \(P(r,\theta)\text{,}\) the rectangular coordinates are determined by \begin{equation*} x=r\cos(\theta) \qquad y=r\sin(\theta) . \end{equation*}

Given the rectangular coordinates \((x,y)\text{,}\) the polar coordinates are determined by \begin{equation*} r^2=x^2+y^2\qquad \tan(\theta) = \frac yx. \end{equation*}

Example9.4.6Converting Between Polar and Rectangular Coordinates
  1. Convert the polar coordinates \(P(2,2\pi/3)\) and \(P(-1,5\pi/4)\) to rectangular coordinates.

  2. Convert the rectangular coordinates \((1,2)\) and \((-1,1)\) to polar coordinates.


Subsection9.4.3Polar Functions and Polar Graphs

Defining a new coordinate system allows us to create a new kind of function, a polar function. Rectangular coordinates lent themselves well to creating functions that related \(x\) and \(y\text{,}\) such as \(y=x^2\text{.}\) Polar coordinates allow us to create functions that relate \(r\) and \(\theta\text{.}\) Normally these functions look like \(r=f(\theta)\text{,}\) although we can create functions of the form \(\theta = f(r)\text{.}\) The following examples introduce us to this concept.

Example9.4.10Introduction to Graphing Polar Functions

Describe the graphs of the following polar functions.

  1. \(r = 1.5\)

  2. \(\theta = \pi/4\)


The basic rectangular equations of the form \(x=h\) and \(y=k\) create vertical and horizontal lines, respectively; the basic polar equations \(r= h\) and \(\theta =\alpha\) create circles and lines through the pole, respectively. With this as a foundation, we can create more complicated polar functions of the form \(r=f(\theta)\text{.}\) The input is an angle; the output is a length, how far in the direction of the angle to go out.

We sketch these functions much like we sketch rectangular and parametric functions: we plot lots of points and “connect the dots” with curves. We demonstrate this in the following example.

Example9.4.12Sketching Polar Functions

Sketch the polar function \(r=1+\cos(\theta)\) on \([0,2\pi]\) by plotting points.

\(\begin{array}{cc} \theta \amp r=1+\cos(\theta) \\ \hline 0 \amp 2 \\ \pi/6 \amp 1.86603 \\ \pi/2 \amp 1 \\ 4\pi/3 \amp 0.5 \\ 7 \pi /4 \amp 1.70711 \\ \end{array}\)

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Table9.4.13Graphing a polar function in Example 9.4.12 by plotting points.

Technology Note: Plotting functions in this way can be tedious, just as it was with rectangular functions. To obtain very accurate graphs, technology is a great aid. Most graphing calculators can plot polar functions; in the menu, set the plotting mode to something like polar or POL, depending on one's calculator. As with plotting parametric functions, the viewing “window” no longer determines the \(x\)-values that are plotted, so additional information needs to be provided. Often with the “window” settings are the settings for the beginning and ending \(\theta\) values (often called \(\theta_{\text{ min } }\) and \(\theta_{\text{ max } }\)) as well as the \(\theta_{\text{ step } }\) — that is, how far apart the \(\theta\) values are spaced. The smaller the \(\theta_{\text{ step } }\) value, the more accurate the graph (which also increases plotting time). Using technology, we graphed the polar function \(r=1+\cos(\theta)\) from Example 9.4.12 in Figure 9.4.14.

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Figure9.4.14Using technology to graph a polar function.
Example9.4.15Sketching Polar Functions

Sketch the polar function \(r=\cos(2\theta)\) on \([0,2\pi]\) by plotting points.


It is sometimes desirable to refer to a graph via a polar equation, and other times by a rectangular equation. Therefore it is necessary to be able to convert between polar and rectangular functions, which we practice in the following example. We will make frequent use of the identities found in Key Idea 9.4.5.

Example9.4.19Converting between rectangular and polar equations.
  1. \(y=x^2\)

  2. \(xy = 1\)

  1. \(\ds r=\frac{2}{\sin(\theta) -\cos(\theta) }\)

  2. \(r=2\cos(\theta)\)


Some curves have very simple polar equations but rather complicated rectangular ones. For instance, the equation \(r=1+\cos(\theta)\) describes a cardiod (a shape important the sensitivity of microphones, among other things; one is graphed in the gallery in the Limaçon section). It's rectangular form is not nearly as simple; it is the implicit equation \(x^4+y^4+2x^2y^2-2xy^2-2x^3-y^2=0\text{.}\) The conversion is not “hard,” but takes several steps, and is left as a problem in the Exercise section.

Gallery of Polar Curves

There are a number of basic and “classic” polar curves, famous for their beauty and/or applicability to the sciences. This section ends with a small gallery of some of these graphs. We encourage the reader to understand how these graphs are formed, and to investigate with technology other types of polar functions.

\newlength{\gallerywidth} \rule{25pt+\marginparwidth+\textwidth}{1pt}


Circles Spiral


Symmetric about \(x\)-axis: \(r=a\pm b\cos(\theta)\text{;}\) Symmetric about \(y\)-axis: \(r=a\pm b\sin(\theta)\text{;}\) \(a,b>0\)

Subsection9.4.5Rose Curves

Symmetric about \(x\)-axis: \(r=a \cos(n\theta)\text{;}\) Symmetric about \(y\)-axis: \(r=a\sin(n\theta)\)

Curve contains \(2n\) petals when \(n\) is even and \(n\) petals when \(n\) is odd.

Special Curves

Earlier we discussed how each point in the plane does not have a unique representation in polar form. This can be a “good” thing, as it allows for the beautiful and interesting curves seen in the preceding gallery. However, it can also be a “bad” thing, as it can be difficult to determine where two curves intersect.

Example9.4.21Finding points of intersection with polar curves

Determine where the graphs of the polar equations \(r=1+3\cos(\theta)\) and \(r=\cos(\theta)\) intersect.


If all one is concerned with is the \((x,y)\) coordinates at which the graphs intersect, much of the above work is extraneous. We know they intersect at \((0,0)\text{;}\) we might not care at what \(\theta\) value. Likewise, using \(\theta =2\pi/3\) and \(\theta=4\pi/3\) can give us the needed rectangular coordinates. However, in the next section we apply calculus concepts to polar functions. When computing the area of a region bounded by polar curves, understanding the nuances of the points of intersection becomes important.


In the following exercises, convert the polar equation to a rectangular equation.

In the following exercises, convert the rectangular equation to a polar equation.

In the following exercises, find the points of intersection of the polar graphs.