18 Polar Coordinates
Learning Objectives
In this section, you will:
- Plot points using polar coordinates.
- Convert from polar coordinates to rectangular coordinates.
- Convert from rectangular coordinates to polar coordinates.
- Transform equations between polar and rectangular forms.
- Identify and graph polar equations by converting to rectangular equations.
Over 12 kilometers from port, a sailboat encounters rough weather and is blown off course by a 16-knot wind (see (Figure 1)). How can the sailor indicate his location to the Coast Guard? In this section, we will investigate a method of representing location that is different from a standard coordinate grid.
![An illustration of a boat on the polar grid.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150014/CNX_Precalc_Figure_08_03_001.jpg)
Plotting Points Using Polar Coordinates
When we think about plotting points in the plane, we usually think of rectangular coordinates in the Cartesian coordinate plane. However, there are other ways of writing a coordinate pair and other types of grid systems. In this section, we introduce polar coordinates, which are points labeled
and plotted on a polar grid. The polar grid is represented as a series of concentric circles radiating out from the pole, or the origin of the coordinate plane.
The polar grid is scaled as the unit circle with the positive x-axis now viewed as the polar axis and the origin as the pole. The first coordinate is the radius or length of the directed line segment from the pole. The angle
measured in radians, indicates the direction of
We move counterclockwise from the polar axis by an angle of
and measure a distance r along a directed line segment in the direction of
Even though we measure
first and then
the polar point is written with the r-coordinate first. For example, to plot the point
we would move
units in the counterclockwise direction and then a length of 2 from the pole. This point is plotted on the grid in (Figure 2).
![Polar grid with point (2, pi/4) plotted.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150040/CNX_Precalc_Figure_08_03_002.jpg)
Plotting a Point on the Polar Grid
Plot the pointon the polar grid.
Show Solution
The angle is found by sweeping in a counterclockwise direction 90° from the polar axis. The point is located at a length of 3 units from the pole in the
direction, as shown in (Figure 3).
![Polar grid with point (3, pi/2) plotted.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150042/CNX_Precalc_Figure_08_03_003.jpg)
Try It
Plot the point in the polar grid.
Show Solution
Plotting a Point in the Polar Coordinate System with a Negative Component
Plot the pointon the polar grid.
Show Solution
We know that is located in the first quadrant. However,
We can approach plotting a point with a negative
in two ways:
- Plot the point
by moving
in the counterclockwise direction and extending a directed line segment 2 units into the first quadrant. Then retrace the directed line segment back through the pole, and continue 2 units into the third quadrant;
- Move
in the counterclockwise direction, and draw the directed line segment from the pole 2 units in the negative direction, into the third quadrant.
See (Figure 4)(a). Compare this to the graph of the polar coordinate shown in (Figure 4)(b).
![Two polar grids. Points (2, pi/6) and (-2, pi/6) are plotted. They are reflections across the origin in Q1 and Q3.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150055/CNX_Precalc_Figure_08_03_005.jpg)
Try It
Plot the points and
on the same polar grid.
Show Solution
Converting from Polar Coordinates to Rectangular Coordinates
When given a set of polar coordinates, we may need to convert them to rectangular coordinates. To do so, we can recall the relationships that exist among the variables and
![Rendered by QuickLaTeX.com \begin{array}{l}\begin{array}{l} \mathrm{cos}\,\theta =\frac{x}{r}\; \text {so } x = r\, \mathrm{cos}\,\theta \end{array}\hfill \\ \mathrm{sin}\,\theta =\frac{y}{r}\; \text {so } y = r\,\mathrm{sin}\,\theta \hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-1dcb45dda9e14c3910c728daef7270a2_l3.png)
Dropping a perpendicular from the point in the plane to the x-axis forms a right triangle, as illustrated in (Figure 5). An easy way to remember the equations above is to think of as the adjacent side over the hypotenuse and
as the opposite side over the hypotenuse.
![Comparison between polar coordinates and rectangular coordinates. There is a right triangle plotted on the x,y axis. The sides are a horizontal line on the x-axis of length x, a vertical line extending from thex-axis to some point in quadrant 1, and a hypotenuse r extending from the origin to that same point in quadrant 1. The vertices are at the origin (0,0), some point along the x-axis at (x,0), and that point in quadrant 1. This last point is (x,y) or (r, theta), depending which system of coordinates you use.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150111/CNX_Precalc_Figure_08_03_007.jpg)
Converting from Polar Coordinates to Rectangular Coordinates
To convert polar coordinates to rectangular coordinates
let
How To
Given polar coordinates, convert to rectangular coordinates.
- Given the polar coordinate
write
and
- Evaluate
and
- Multiply
b y
to find the x-coordinate of the rectangular form.
- Multiply
by
to find the y-coordinate of the rectangular form.
Writing Polar Coordinates as Rectangular Coordinates
Write the polar coordinates as rectangular coordinates.
Show Solution
Use the equivalent relationships.
![Rendered by QuickLaTeX.com \begin{array}{l}\begin{array}{l}\\ x=r\, \mathrm{cos}\,\theta \end{array}\hfill \\ x=3\mathrm{cos}\,\frac{\pi }{2}=0\hfill \\ y=r\, \mathrm{sin}\,\theta \hfill \\ y=3\mathrm{sin}\,\frac{\pi }{2}=3\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-5b0829ec7e35d91f61cc00554e584666_l3.png)
The rectangular coordinates are See (Figure 6).
![Illustration of (3, pi/2) in polar coordinates and (0,3) in rectangular coordinates - they are the same point!](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150122/CNX_Precalc_Figure_08_03_008.jpg)
Writing Polar Coordinates as Rectangular Coordinates
Write the polar coordinates as rectangular coordinates.
Show Solution
See (Figure 7). Writing the polar coordinates as rectangular, we have
![Rendered by QuickLaTeX.com \begin{array}{l}x=r\,\, \mathrm{cos}\,\theta \hfill \\ x=-2\mathrm{cos}\left(0\right)=-2\hfill \\ \hfill \\ y=r\, \mathrm{sin}\,\theta \hfill \\ y=-2\mathrm{sin}\left(0\right)=0\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-790b22f78dd4d72ea4b12d954dfc252e_l3.png)
The rectangular coordinates are also
![Illustration of (-2, 0) in polar coordinates and (-2,0) in rectangular coordinates - they are the same point!](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150129/CNX_Precalc_Figure_08_03_009.jpg)
Try It
Write the polar coordinates as rectangular coordinates.
Show Solution
Converting from Rectangular Coordinates to Polar Coordinates
To convert rectangular coordinates to polar coordinates, we will use two other familiar relationships. With this conversion, however, we need to be aware that a set of rectangular coordinates will yield more than one polar point.
Converting from Rectangular Coordinates to Polar Coordinates
Converting from rectangular coordinates to polar coordinates requires the use of one or more of the relationships illustrated in (Figure 8).
![Rendered by QuickLaTeX.com \begin{array}{l}\mathrm{cos}\,\theta =\frac{x}{r}\,\,\text{ or}\,\,\,x = r\, \mathrm{cos}\,\theta \hfill \\ \mathrm{sin}\,\theta =\frac{y}{r}\,\,\text{ or}\,\,\,y = r\, \mathrm{sin}\,\theta \hfill \\ \,\,\,\,\,\,{r}^{2}={x}^{2}+{y}^{2}\hfill \\ \mathrm{tan}\,\theta =\frac{y}{x}\,\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-d719a17e5b04886951061753ca6711f0_l3.png)
![](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150144/CNX_Precalc_Figure_08_03_010new.jpg)
Writing Rectangular Coordinates as Polar Coordinates
Convert the rectangular coordinates to polar coordinates.
Show Solution
We see that the original point is in the first quadrant. To find
use the formula
This gives
![Rendered by QuickLaTeX.com \begin{array}{l}\,\,\,\,\,\,\,\,\,\,\,\mathrm{tan}\,\theta =\frac{3}{3}\hfill \\ \,\,\,\,\,\,\,\,\,\,\,\mathrm{tan}\,\theta =1\hfill \\ \,\,\,{\mathrm{tan}}^{-1}\left(1\right)=\frac{\pi }{4}\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-608e938c6560e8fcb50fde36306dbfb3_l3.png)
To find we substitute the values for
and
into the formula
We know that
must be positive, as
is in the first quadrant. Thus
![Rendered by QuickLaTeX.com \begin{array}{l}\begin{array}{l}\\ r=\sqrt{{3}^{2}+{3}^{2}}\end{array}\hfill \\ r=\sqrt{9+9}\hfill \\ r=\sqrt{18}=3\sqrt{2}\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-bf7b1f3c73a05229771b603468be51e8_l3.png)
So, and
giving us the polar point
See (Figure 9).
![Illustration of (3rad2, pi/4) in polar coordinates and (3,3) in rectangular coordinates - they are the same point!](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150158/CNX_Precalc_Figure_08_03_011.jpg)
Analysis
There are other sets of polar coordinates that will be the same as our first solution. For example, the points and
will coincide with the original solution of
The point
indicates a move further counterclockwise by
which is directly opposite
The radius is expressed as
However, the angle
is located in the third quadrant and, as
is negative, we extend the directed line segment in the opposite direction, into the first quadrant. This is the same point as
The point
is a move further clockwise by
from
The radius,
is the same.
Transforming Equations between Polar and Rectangular Forms
We can now convert coordinates between polar and rectangular form. Converting equations can be more difficult, but it can be beneficial to be able to convert between the two forms. Since there are a number of polar equations that cannot be expressed clearly in Cartesian form, and vice versa, we can use the same procedures we used to convert points between the coordinate systems. We can then use a graphing calculator to graph either the rectangular form or the polar form of the equation.
How To
Given an equation in polar form, graph it using a graphing calculator.
- Change the MODE to POL, representing polar form.
- Press the Y= button to bring up a screen allowing the input of six equations:
- Enter the polar equation, set equal to
- Press GRAPH.
Writing a Cartesian Equation in Polar Form
Write the Cartesian equation in polar form.
Show Solution
The goal is to eliminate and
from the equation and introduce
and
Ideally, we would write the equation
as a function of
To obtain the polar form, we will use the relationships between
and
Since
and
we can substitute and solve for
![Rendered by QuickLaTeX.com \begin{array}{ll}\text{ }{\left(r\,\mathrm{cos}\,\theta \right)}^{2}+{\left(r\,\mathrm{sin}\,\theta \right)}^{2}=9\hfill & \hfill \\ \text{ }{r}^{2}{\mathrm{cos}}^{2}\theta +{r}^{2}{\mathrm{sin}}^{2}\theta =9\hfill & \hfill \\ \text{ }{r}^{2}\left({\mathrm{cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta \right)=9\hfill & \hfill \\ \,\text{ }{r}^{2}\left(1\right)=9 \hfill & {\text{Substitute cos}}^{2}\theta +{\mathrm{sin}}^{2}\theta =1.\hfill \\ \text{ }r=±3\begin{array}{cccc}& & & \end{array}\hfill & \text{Use the square root property}.\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-c6a6b65308c334ed29e83ba66a738f66_l3.png)
Thus, and
should generate the same graph. See (Figure 10).
![Plotting a circle of radius 3 with center at the origin in polar and rectangular coordinates. It is the same in both systems.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150200/CNX_Precalc_Figure_08_03_016.jpg)
![Rendered by QuickLaTeX.com \,{x}^{2}+{y}^{2}=9\,](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-d6cee199c398facc0648762e0ef96c86_l3.png)
![Rendered by QuickLaTeX.com \,r=3](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-7ca44f7ff1cdbd65502a9b26e30c53a8_l3.png)
To graph a circle in rectangular form, we must first solve for
![Rendered by QuickLaTeX.com \begin{array}{l}\begin{array}{l}\\ {x}^{2}+{y}^{2}=9\end{array}\hfill \\ \text{ }{y}^{2}=9-{x}^{2}\hfill \\ \text{ }y=±\sqrt{9-{x}^{2}}\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-6805514b730b162970fdebd615912da3_l3.png)
Note that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the form and
Press GRAPH.
Rewriting a Cartesian Equation as a Polar Equation
Rewrite the Cartesian equation as a polar equation.
Show Solution
This equation appears similar to the previous example, but it requires different steps to convert the equation.
We can still follow the same procedures we have already learned and make the following substitutions:
![Rendered by QuickLaTeX.com \begin{array}{ll}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r}^{2}=6y\hfill & \text{Use }{x}^{2}+{y}^{2}={r}^{2}.\hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{r}^{2}=6r\, \mathrm{sin}\,\theta \hfill & \text{Substitute}\,y=r\mathrm{sin}\,\theta .\hfill \\ \text{ }{r}^{2}-6r\, \mathrm{sin}\,\theta =0\hfill & \text{Set equal to 0}.\hfill \\ \text{ }r\left(r-6\mathrm{sin}\,\theta \right)=0\hfill & \text{Factor and solve}.\hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,r=0\hfill & \text{We reject }r=0,\,\text{as it only represents one point, }\left(0,0\right).\hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{or}\,r = 6\mathrm{sin}\,\theta \begin{array}{cccc}& & & \end{array}\hfill & \hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-e738740fc9626e7f10531450a86959ff_l3.png)
Therefore, the equations and
gives us the same graph. See (Figure).
![Rendered by QuickLaTeX.com \, {x}^{2}+ {y}^{2}= 6y](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-80ad27d1b80aa5d078b8d2687677fe35_l3.png)
![Rendered by QuickLaTeX.com \, r=6\mathrm{sin}\,\theta](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-86e450425edbfffc68593728264c91ba_l3.png)
The Cartesian or rectangular equation is plotted on the rectangular grid, and the polar equation is plotted on the polar grid. Clearly, the graphs are identical.
Rewriting a Cartesian Equation in Polar Form
Rewrite the Cartesian equationas a polar equation.
Show Solution
We will use the relationships and
![Rendered by QuickLaTeX.com \begin{array}{ll}\text{ }y = 3x+2\hfill & \hfill \\ \text{ }r\,\mathrm{sin}\,\theta =3r\, \mathrm{cos}\,\theta +2\hfill & \hfill \\ \,r \mathrm{sin}\,\theta -3r\, \mathrm{cos}\,\theta =2\hfill & \hfill \\ r\, \left(\mathrm{sin}\,\theta -3\mathrm{cos}\,\theta \right)=2\hfill & \text{Isolate }r.\hfill \\ \text{ }\,\,\,r=\frac{2}{\mathrm{sin}\,\theta -3\mathrm{cos}\,\theta }\begin{array}{cccc}& & & \end{array}\hfill & \text{Solve for }r.\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-0f54ba03655b6333ba92189d1fc24488_l3.png)
Try It
Rewrite the Cartesian equation in polar form.
Show Solution
Identify and Graph Polar Equations by Converting to Rectangular Equations
We have learned how to convert rectangular coordinates to polar coordinates, and we have seen that the points are indeed the same. We have also transformed polar equations to rectangular equations and vice versa. Now we will demonstrate that their graphs, while drawn on different grids, are identical.
Graphing a Polar Equation by Converting to a Rectangular Equation
Covert the polar equation to a rectangular equation, and draw its corresponding graph.
Show Solution
The conversion is
![Rendered by QuickLaTeX.com \begin{array}{l}\text{ }r=2\mathrm{sec}\,\theta \hfill \\ \,\,\,\,\text{ }r=\frac{2}{\mathrm{cos}\,\theta }\hfill \\ r\mathrm{cos}\,\theta =2\hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x=2\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-45a9c872a8ff5a7314e72f00702348ed_l3.png)
Notice that the equation drawn on the polar grid is clearly the same as the vertical line
drawn on the rectangular grid (see (Figure 12)). Just as
is the standard form for a vertical line in rectangular form,
is the standard form for a vertical line in polar form.
![Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are lines.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150210/CNX_Precalc_Figure_08_03_013.jpg)
A similar discussion would demonstrate that the graph of the function will be the horizontal line
In fact,
is the standard form for a horizontal line in polar form, corresponding to the rectangular form
Rewriting a Polar Equation in Cartesian Form
Rewrite the polar equation as a Cartesian equation.
Show Solution
The goal is to eliminate and
and introduce
and
We clear the fraction, and then use substitution. In order to replace
with
and
we must use the expression
![Rendered by QuickLaTeX.com \begin{array}{llll}\text{ }r=\frac{3}{1-2\mathrm{cos}\,\theta }\hfill & \hfill & \hfill & \hfill \\ r\left(1-2\mathrm{cos}\,\theta \right)=3\hfill & \hfill & \hfill & \hfill \\ \,\,\,\,\,\,r\left(1-2\left(\frac{x}{r}\right)\right)=3\hfill & \hfill & \hfill & \text{Use }\mathrm{cos}\,\theta =\frac{x}{r}\text{ to eliminate }\theta .\hfill \\ \text{ }\,\,\,r-2x=3\hfill & \hfill & \hfill & \hfill \\ \text{ }r=3+2x\hfill & \hfill & \hfill & \text{Isolate }r.\hfill \\ \text{ }{r}^{2}={\left(3+2x\right)}^{2}\hfill & \hfill & \hfill & \text{Square both sides}.\hfill \\ \text{ }\,{x}^{2}+{y}^{2}={\left(3+2x\right)}^{2}\hfill & \hfill & \hfill & \text{Use }{x}^{2}+{y}^{2}={r}^{2}.\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-2a8d8edbbb01dc737762767aad68d9bf_l3.png)
The Cartesian equation is However, to graph it, especially using a graphing calculator or computer program, we want to isolate
![Rendered by QuickLaTeX.com \begin{array}{l}{x}^{2}+{y}^{2}={\left(3+2x\right)}^{2}\hfill \\ \text{ }{y}^{2}={\left(3+2x\right)}^{2}-{x}^{2}\hfill \\ \text{ }y=±\sqrt{{\left(3+2x\right)}^{2}-{x}^{2}}\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-fd24789a6b2f5f0629c2c92115e1fe78_l3.png)
When our entire equation has been changed from and
to
and
we can stop, unless asked to solve for
or simplify. See (Figure 13).
![Plots of the equations stated above - the plots are the same in both rectangular and polar coordinates. They are hyperbolas.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150212/CNX_Precalc_Figure_08_03_015.jpg)
The “hour-glass” shape of the graph is called a hyperbola. Hyperbolas have many interesting geometric features and applications, which can be investigated further in Analytic Geometry.
Analysis
In this example, the right side of the equation can be expanded and the equation simplified further, as shown above. However, the equation cannot be written as a single function in Cartesian form. We may wish to write the rectangular equation in the hyperbola’s standard form. To do this, we can start with the initial equation.
![Rendered by QuickLaTeX.com \begin{array}{ll}\,\text{ }{x}^{2}+{y}^{2}={\left(3+2x\right)}^{2}\hfill & \hfill \\ \,\,\text{ }{x}^{2}+{y}^{2}-{\left(3+2x\right)}^{2}=0\hfill & \hfill \\ {x}^{2}+{y}^{2}-\left(9+12x+4{x}^{2}\right)=0\hfill & \hfill \\ \,\text{ }{x}^{2}+{y}^{2}-9-12x-4{x}^{2}=0\hfill & \hfill \\ \,\text{ }-3{x}^{2}-12x+{y}^{2}=9\hfill & \text{Multiply through by }-1.\hfill \\ \,\,\text{ }\,3{x}^{2}+12x-{y}^{2}=-9\hfill & \hfill \\ \,\,\text{ }\,3\left({x}^{2}+4x+\,\,\,\,\,\,\,\,\,\right)-{y}^{2}=-9\hfill & \text{Organize terms to complete the square for}\,x.\hfill \\ \,\,\text{ }\,3\left({x}^{2}+4x+4\right)-{y}^{2}=-9+12\hfill & \hfill \\ \text{ }3{\left(x+2\right)}^{2}-{y}^{2}=3\hfill & \hfill \\ \,\text{ }{\left(x+2\right)}^{2}-\frac{{y}^{2}}{3}=1\hfill & \hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-5a046075b732d967bb2ae6d8c41719f9_l3.png)
Try It
Rewrite the polar equation in Cartesian form.
Show Solution
or, in the standard form for a circle,
Rewriting a Polar Equation in Cartesian Form
Rewrite the polar equationin Cartesian form.
Show Solution
![Rendered by QuickLaTeX.com \begin{array}{ll}\text{ }r=\mathrm{sin}\left(2\theta \right)\hfill & \text{Use the double angle identity for sine}.\hfill \\ \text{ }r=2\mathrm{sin}\,\theta \mathrm{cos}\,\theta \begin{array}{cccc}& & & \end{array}\hfill & \text{Use }\mathrm{cos}\,\theta =\frac{x}{r}\text{ and }\mathrm{sin}\,\theta =\frac{y}{r}.\hfill \\ \text{ }r=2\left(\frac{x}{r}\right)\left(\frac{y}{r}\right)\hfill & \text{Simplify}.\hfill \\ \text{ }r=\frac{2xy}{{r}^{2}}\hfill & \text{ Multiply both sides by }{r}^{2}.\hfill \\ \text{ }{r}^{3}=2xy\hfill & \hfill \\ {\left(\sqrt{{x}^{2}+{y}^{2}}\right)}^{3}=2xy\hfill & \text{As}\,{x}^{2}+{y}^{2}={r}^{2},r=\sqrt{{x}^{2}+{y}^{2}}.\hfill \end{array}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-eb35a4d45f5d599dde06950f5e2d13b4_l3.png)
This equation can also be written as
![Rendered by QuickLaTeX.com {\left({x}^{2}+{y}^{2}\right)}^{\frac{3}{2}}=2xy\,\text{or}\,{x}^{2}+{y}^{2}={\left(2xy\right)}^{\frac{2}{3}}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-c34cccb465bfde699764024946a30ec5_l3.png)
Access these online resources for additional instruction and practice with polar coordinates.
Key Equations
Conversion formulas | ![]() |
Key Concepts
- The polar grid is represented as a series of concentric circles radiating out from the pole, or origin.
- To plot a point in the form
move in a counterclockwise direction from the polar axis by an angle of
and then extend a directed line segment from the pole the length of
in the direction of
If
is negative, move in a clockwise direction, and extend a directed line segment the length of
in the direction of
See (Figure 3).
- If
is negative, extend the directed line segment in the opposite direction of
See (Figure 4).
- To convert from polar coordinates to rectangular coordinates, use the formulas
and
See (Figure 6) and (Figure 7).
- To convert from rectangular coordinates to polar coordinates, use one or more of the formulas:
and
See (Figure 9).
- Transforming equations between polar and rectangular forms means making the appropriate substitutions based on the available formulas, together with algebraic manipulations. See (Figure 10), (Figure 11), and (Figure).
- Using the appropriate substitutions makes it possible to rewrite a polar equation as a rectangular equation, and then graph it in the rectangular plane. See (Figure 12), (Figure 13), and (Figure).
Section Exercises
Verbal
How are polar coordinates different from rectangular coordinates?
Show Solution
For polar coordinates, the point in the plane depends on the angle from the positive x-axis and distance from the origin, while in Cartesian coordinates, the point represents the horizontal and vertical distances from the origin. For each point in the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations.
How are the polar axes different from the x– and y-axes of the Cartesian plane?
Explain how polar coordinates are graphed.
Show Solution
Determine for the point, then move
units from the pole to plot the point. If
is negative, move
units from the pole in the opposite direction but along the same angle. The point is a distance of
away from the origin at an angle of
from the polar axis.
How are the points latex]\,\left(3,\frac{\pi }{2}\right)\,[/latex] and in polar coordinates related?
Explain why the points and
are the same in polar coordinates.
Show Solution
The point has a positive angle but a negative radius and is plotted by moving to an angle of
and then moving 3 units in the negative direction. This places the point 3 units down the negative y-axis. The point
has a negative angle and a positive radius and is plotted by first moving to an angle of
and then moving 3 units down, which is the positive direction for a negative angle. The point is also 3 units down the negative y-axis.
Algebraic
For the following exercises, convert the given polar coordinates to Cartesian coordinates with and
Remember to consider the quadrant in which the given point is located when determining
for the point.
Show Solution
![Rendered by QuickLaTeX.com \left(6,-\frac{\pi }{4}\right)](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-14df8b1ebb395e830a6e1058fa860a76_l3.png)
Show Solution
For the following exercises, convert the given Cartesian coordinates to polar coordinates with Remember to consider the quadrant in which the given point is located.
Show Solution
![Rendered by QuickLaTeX.com \left(-4,6\right)](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-46dcd9d3898bf55c3e7c3af73128aeca_l3.png)
Show Solution
Show Solution
For the following exercises, convert the given Cartesian equation to a polar equation.
Show Solution
Show Solution
![Rendered by QuickLaTeX.com {x}^{2}+{y}^{2}=3x](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-1e95de5ecb891db7e93943592ffd1e75_l3.png)
Show Solution
Show Solution
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Show Solution
For the following exercises, convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.
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or
circle
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line
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line
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hyperbola
![Rendered by QuickLaTeX.com r=4](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-b14f8d37554b47394ab123bfa86ff407_l3.png)
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circle
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line
Graphical
For the following exercises, find the polar coordinates of the point.
![Polar coordinate system with a point located on the third concentric circle and pi/2.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150215/CNX_Precalc_Figure_08_03_201n.jpg)
![Polar coordinate system with a point located on the third concentric circle and midway between pi/2 and pi in the second quadrant.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150230/CNX_Precalc_Figure_08_03_202n.jpg)
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![Polar coordinate system with a point located midway between the first and second concentric circles and a third of the way between pi and 3pi/2 (closer to pi).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150232/CNX_Precalc_Figure_08_03_203n.jpg)
![Polar coordinate system with a point located on the fifth concentric circle and pi.](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150243/CNX_Precalc_Figure_08_03_204n.jpg)
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![Polar coordinate system with a point located on the fourth concentric circle and a third of the way between 3pi/2 and 2pi (closer to 3pi/2).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/3252/2018/07/19150257/CNX_Precalc_Figure_08_03_205n.jpg)
For the following exercises, plot the polar points.
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![Rendered by QuickLaTeX.com \left(3.5,\frac{7\pi }{4}\right)](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-9ab28cd328eccdc3ec790e7743476288_l3.png)
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For the following exercises, convert the equation from rectangular to polar form and graph on the polar axis.
![Rendered by QuickLaTeX.com 5x-y=6](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-a0d22b5c442e3f00ad6c0a71da54317d_l3.png)
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For the following exercises, convert the equation from polar to rectangular form and graph on the rectangular plane.
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![Rendered by QuickLaTeX.com \theta =-\frac{2\pi }{3}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-ee1db5ac1f163e23fa813edb1bfd6297_l3.png)
![Rendered by QuickLaTeX.com \theta =\frac{\pi }{4}](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-b2282bb0a1e3c93a6810d5da070e4556_l3.png)
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Technology
Use a graphing calculator to find the rectangular coordinates of Round to the nearest thousandth.
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Use a graphing calculator to find the rectangular coordinates ofRound to the nearest thousandth.
Use a graphing calculator to find the polar coordinates ofin degrees. Round to the nearest thousandth.
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Use a graphing calculator to find the polar coordinates of in degrees. Round to the nearest hundredth.
Use a graphing calculator to find the polar coordinates of in radians. Round to the nearest hundredth.
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Extensions
Describe the graph of
Describe the graph of
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A vertical line with units left of the y-axis.
Describe the graph of
Describe the graph of
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A horizontal line with units below the x-axis.
What polar equations will give an oblique line?
For the following exercise, graph the polar inequality.
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![Rendered by QuickLaTeX.com \theta =\frac{\pi }{4},\,r\,\ge -3](https://university.pressbooks.pub/app/uploads/quicklatex/quicklatex.com-1fb8643cc94f9d7ebc24cd5f3015cb20_l3.png)
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Glossary
- polar axis
- on the polar grid, the equivalent of the positive x-axis on the rectangular grid
- polar coordinates
- on the polar grid, the coordinates of a point labeled
where
indicates the angle of rotation from the polar axis and
represents the radius, or the distance of the point from the pole in the direction of
- pole
- the origin of the polar grid