(Note that we talk about converting back and forth from Polar Complex Form to Rectangular Complex form here in the Trigonometry and the Complex Plane section. Also note that we discussed Parametric Equations here, which may seem similar to Polar Equations, since they both have applications in Trigonometry.)
Plotting Points Using Polar Coordinates
So far, we’ve plotted points using rectangular (or Cartesian) coordinates, since the points since we are going back and forth $ x$ units, and up and down $ y$ units. In the Polar Coordinate System, we go around the origin or the pole a certain distance out, and a certain angle from the positive $ x$–axis:
The ordered pairs, called polar coordinates, are in the form $ \left( {r,\theta } \right)$, with $ r>0$ being the number of units from the origin or pole, like a radius of a circle, and $ \theta $ being the angle (in degrees or radians) formed by the ray on the positive $ x$–axis (polar axis), going counter-clockwise. If $ r<0$, the point is $ r$ units in the opposite direction (across the origin or pole) of the angle $ \theta $. If $ \theta <0$, you go clockwise with the angle, starting with the positive $ x$–axis, instead of counter-clockwise.
To plot a point, you typically circle around the positive $ x$–axis $ \theta $ degrees first, and then go out from the origin or pole $ r$ units. If $ r$ is negative, you go in the oppositive direction (180°) $ r$ units. If both $ r$ and the angle are negative, you have to make sure you go clockwise to get the angle, but in the opposite direction $ r$ units.
Here’s a polar graph with some points on it; note that we typically count in increments of 15°, or $ \displaystyle \frac{\pi }{{12}}$:
For a point $ \left( {r,\theta } \right)$, do you see how you always go counter-clockwise (or clockwise, if you have a negative angle) until you reach the angle you want, and then out from the center $ r$ units, if $ r$ is positive? Again, if $ r$ is negative, you go in the opposite direction from the angle $ r$ units. If both $ r$ and the angle $ \theta $ are negative, you have to make sure you go clockwise to get the angle, but in the opposite direction $ r$ units.
You may be asked to rename a point in several different ways, for example, between $ \left[ {-2\pi ,2\pi } \right)$ or $ \left[ {-360{}^\circ ,360{}^\circ } \right)$. For example, if we wanted to rename the point $ \left( {6,240{}^\circ } \right)$ three other different ways between $ \left[ {-360{}^\circ ,360{}^\circ } \right)$, by looking at the graph above, we’d get $ \left( {6,-120{}^\circ } \right)$ (subtract 360°), $ \left( {-6,60{}^\circ } \right)$ (make $ r$ negative and subtract 180°), and $ \left( {-6,-300{}^\circ } \right)$(subtract another 360°). (Remember that 240° and –120°, and 60° and –300° are co-terminal angles). To get these, if the first number ($ r$) is negative, go in the opposite direction, and if the angle is negative, go clockwise instead of counterclockwise from the positive $ x$–axis. Note that $ \left( {r,\theta } \right)$ and $ \left( {-r,-\theta } \right)$ aren’t the same point!
Polar-Rectangular Point Conversions
You will probably be asked to convert coordinates between polar form and rectangular form.
Converting from Polar to Rectangular Coordinates
Let’s first convert from polar to rectangular form; to do this we use the following formulas, using Right Triangle Trigonometry:
This conversion is pretty straight-forward; examples are below.
Converting from Rectangular to Polar Coordinates
Converting from rectangular coordinates to polar coordinates can be a little trickier since we need to check the quadrant of the rectangular point to get the correct angle; the quadrants must match. Here are the formulas:
$ \displaystyle r=\sqrt{{{{x}^{2}}+{{y}^{2}}}}\,\,\,\,\,\text{(this will be positive)}$
$ \displaystyle \theta ={{\tan }^{{-1}}}\left( {\frac{y}{x}} \right)\,\,\text{(check for correct quadrant)}$
One word of caution: If $ x=0$, we’ll get an error when trying to obtain $ \theta $. In these cases, graph the point to get the angle; it will either be $ \displaystyle \frac{\pi }{2}\,\,(90{}^\circ )$ or $ \displaystyle \frac{{3\pi }}{2}\,\,(270{}^\circ )$, depending on where the point lies on the $ y$-axis.
Again, we need to check quadrants when using the calculator to get $ {{\tan }^{{-1}}}$. We’ll have to add the following degrees or radians when the point is in the following quadrants; this is because the $ {{\tan }^{{-1}}}$ function on the calculator only gives answers back in the interval $ \displaystyle \left( {-\frac{\pi }{2},\frac{\pi }{2}} \right)$), as shown in the Inverse Trigonometric Functions section:
Note that there can be multiple “answers” when converting from rectangular to polar, since polar points can be represented in many different ways (co-terminal angles, positive or negative “$ r$”, and so on). Thus, it is typically easier to convert from polar to rectangular.
Examples
Here are some examples of conversions both way; note you may be asked to convert back to polar into degrees or radians. For converting back to polar, make sure answers are either between 0 and 360° for degrees or 0 to $ 2\pi $ for radians. (And again, note that when we convert back to polar coordinates, we may not always get the same representation of the polar point we started out with.)
Here are some problems going from Rectangular to Polar without special angles (angles found on the Unit Circle). Note that we may have to add $ \pi $ or $ 2\pi $ to our answer, depending on which Quadrant the point lies in:
Angle Conversions in Calculator
Note that you can also use “2nd apps (angle)” on your graphing calculator to do these conversions, but you won’t get the answers with the roots in them (you’ll get decimals that aren’t “exact”). You have to solve for the $ x$ and $ y$, or $ r$ and $ \theta $ separately, using the “,” above the 7 for the comma. Make sure you have your calculator either in DEGREES or RADIANS (in MODE), depending on what you’re working with. Here are some examples:
Converting Equations from Rectangular to Polar
To convert Rectangular Equations (Cartesian Coordinates) to Polar Equations, we want to get rid of the $ x$’s and $ y$’s and only have $ r$’s and/or $ \theta $’s in the answer. We do this with the following equations:
$ \begin{array}{l}x=r\cos \,\theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{x}^{2}}+{{y}^{2}}={{r}^{2}}\\y=r\,\sin \,\theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1\end{array}$
Here are some examples; note that we want to solve for $ r$ if we can; in the case of quadratics or higher degrees, this may involve moving everything to one side and factoring.
Also note that when we also get $ r=0$ (the pole) for the answers, this is one point only, and in these cases, the pole is included in the other part of the answers. Thus, we can typically discard $ r=0$.
Converting Equations from Polar to Rectangular
To convert Polar Equations to Rectangular Equations (Cartesian Coordinates), we want to get rid of the $ r$’s and $ \theta $’s and only have $ x$’s and/or $ y$’s in the answer. We do this with the following equations, depending on what we have in the polar equation:
$ \displaystyle r=\sqrt{{{{x}^{2}}+\,{{y}^{2}}}}\,\,\,\,\,\,\,\,\cos \theta =\frac{x}{r}=\frac{x}{{\sqrt{{{{x}^{2}}+{{y}^{2}}}}}}$
$ \displaystyle \theta ={{\tan }^{{-1}}}\left( {\frac{y}{x}} \right)\,\,\,\,\,\sin \theta =\frac{y}{r}=\frac{y}{{\sqrt{{{{x}^{2}}+{{y}^{2}}}}}}$
Here are some examples. Note that sometimes we may be asked to Complete the Square to get the equation in a circle (or other conic) form; we learned how to do this in the Factoring Quadratics and Completing the Square section here.
Drawing Polar Graphs
I find that drawing polar graphs is a combination of part memorizing and part knowing how to create polar t-charts. Below are tables of some of the more common polar graphs, including t-charts in both degrees and radians.
Note that you can also put these in your graphing calculator, using radians or degrees. Here is an example with radians: mode: RADIAN, POLAR (instead of FUNCTION). Then, in window, set: θ = [0, 2π], θstep = π/24 or .1, X = [–10, 10], Y = [–6, 6]. Then using “Y=” to put in the equation, using 
for theta. You can also use 2nd window (tblset) to set up the start and delta points in a table, and 2nd graph (table) to see the points.
You can also just set the mode to POLAR, put in the graph, and use zoom, ZStandard (option 6) and then zoom, ZTrig (option 7) to set the window (also try zoom, ZoomFit (option 0) if it doesn’t look right). By putting in smaller values of θstep, the graph is drawn more slowly and more accurately; to redraw graph, you can turn the graph off and back on by going to “=” and un-highlighting and highlighting it back again before hitting “graph”. Here is the result:
Polar Graphs
Let’s start with polar equations that result in circle graphs:
Here are some polar equations that result in lines:
Here are graphs that we call Cardioids and Limacons. They are in the form $ r=a+b\cos \theta $ or $ r=a+b\sin \theta $, and $ b$ can be positive or negative. Note that, unlike their rectangular equivalents, $ r=a+b\cos \theta $ and $ r=-a+b\cos \theta $ (same with sin) are the same polar graph! Try it!
First, the Cardioids (Hearts); note that these and the Limecon “Loops” touch the pole (origin), while the Limecon “Beans” do not:
Here are the Limacons:
Graphs of Roses produce “petals” and are in the form $ r=a\cos \left( {b\theta } \right)$ or $ r=a\sin \left( {b\theta } \right)$. Note that since we have the starting point for these graphs, and the distance between the petals, the t-chart isn’t that helpful. The t-charts do tell us the order in which the petals are drawn; they are drawn in loops like figure eights. (In the t-charts, I made the $ \Delta $ angle the same as the distance between petals).
Let’s start with the cosine Rose graphs:
And here are some sine Rose graphs:
Note: For a rose graph, you may be asked to name the order that petals are drawn. One way to do this is to use the angle measurements $ \displaystyle 0,\,\frac{\pi }{4},\,\frac{{3\pi }}{4},\,\frac{{5\pi }}{4},\,\frac{{7\pi }}{4}$, solve for $ r$, and observe the order of the petals. You can also use the graphing calculator as shown above, but make the θstep smaller to slow down the drawing of the graph.
Note: You may also see a combination of a rose and a limacon in the form $ r=a+b\cos \left( {k\theta } \right),\,\,r=a+b\sin \left( {k\theta } \right),\,\,k>1$. In these cases, you may see graphs that don’t meet at the origin; try these on your calculator!
Here are a couple more polar graphs (Spirals and Lemniscates) that you might see:
You might be asked to obtain the equation of a polar function from a graph:
Here are more types of questions you may get asked when studying polar graphs:
Polar Graph Points of Intersection
To find the intersection points for sets of polar curves, it’s helpful to draw the curves and also to solve algebraically. To solve algebraically, we just set the $ r$’s together and solve for $ \theta $. Note also that after we solve for one variable (like $ \theta $), we have to plug it back in either equation to get the other coordinate (like $ r$).
We also have to be careful since there are “phantom” or “elusive” points, typically at the pole. The reason these points are “phantom” is because, although we don’t necessarily get them algebraically, we can see them on a graph. This is because, with an “$ r$” of 0, the $ \theta $ could really be anything, since we aren’t going out any distance. We will also see phantom points when one of the equations is “$ r=$ constant”, since another way to write this is “$ r=$ the negative of that constant”. Note that with “phantom” points, both equations do not have to work; I know, it’s weird. To get all these elusive points, you put in the $ r$ value in both curves to see what additional points you get.
Find the intersection points for the following sets of polar curves (algebraically) and also draw a sketch. Find the intersections when $ \theta $ is between 0 and $ \boldsymbol {2\pi} $, and $ r$ is positive.
It also might be good to know the sequence in which the polar graphs are drawn; in other words, from 0 to $ 2\pi $, which parts of the graphs are drawn before the other graphs. (Check it out on a graphing calculator, where you can see it!) You can use a t-chart, or set the polar equation to 0 if the graph crosses the pole, and test points in between. Here are some examples:
Learn these rules, and practice, practice, practice!
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