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Mathematics LibreTexts

8.2 Polar Coordinates

The coordinate system we are most familiar with is called the Cartesian coordinate system, a rectangular plane divided into four quadrants by the horizontal and vertical axes.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.jpg

 

x

In earlier chapters, we often found the Cartesian coordinates of a point on a circle at a given angle from the positive horizontal axis.  Sometimes that angle, along with the point’s distance from the origin, provides a more useful way of describing the point’s location than conventional Cartesian coordinates.

Polar Coordinates

Polar coordinates of a point consist of an ordered pair, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif, where r is the distance from the point to the origin, and θ is the angle measured in standard position.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image005.gif

 

Notice that if we were to “grid” the plane for polar coordinates, it would look like the graph to the right, with circles at incremental radii, and rays drawn at incremental angles. 

 

 

Example 1

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image006.gifPlot the polar point File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif.

 

This point will be a distance of 3 from the origin, at an angle of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image010.gif.  Plotting this,

 

 

Example 2

Plot the polar point File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image012.gif.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image013.gifTypically we use positive r values, but occasionally we run into cases where r is negative.  On a regular number line, we measure positive values to the right and negative values to the left.  We will plot this point similarly.  To start we rotate to an angle of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image015.gif.  Moving this direction, into the first quadrant, would be positive r values.  For negative r values, we move the opposite direction, into the third quadrant.  Plotting this:

Note the resulting point is the same as the polar point File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image017.gif.

 

 

Try it Now

  1. Plot the following points given in polar coordinates and label them.

a. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image019.gif            b. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image021.gif           c. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image023.gif

 

 

Converting Points

To convert between polar coordinates and Cartesian coordinates, we recall the relationships we developed back in Chapter 5.

 

 

Converting Between Polar and Cartesian Coordinates

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image024.gifTo convert between polar File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gifand Cartesian (x, y) coordinates, we use the relationships

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image026.gif               File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image028.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image030.gif                File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image032.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image034.gif               File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif

 

 

From these relationship and our knowledge of the unit circle, if r = 1 and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image038.gif, the polar coordinates would be File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image040.gif, and the corresponding Cartesian coordinatesFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image042.gif.

 

Remembering your unit circle values will come in very handy as you convert between Cartesian and polar coordinates.

 

 

 

 

 

 

 

Example 3

Find the Cartesian coordinates of a point with polar coordinates File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image044.gif.

 

To find the x and y coordinates of the point,

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image046.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image048.gif

 

The Cartesian coordinates are File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image050.gif.

 

 

Example 4

Find the polar coordinates of the point with Cartesian coordinates File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image052.gif.

 

We begin by finding the distance r using the Pythagorean relationship File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image054.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image056.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image058.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image060.gif

 

Now that we know the radius, we can find the angle using any of the three trig relationships.  Keep in mind that any of the relationships will produce two solutions on the circle, and we need to consider the quadrant to determine which solution to accept.  Using the cosine, for example:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image062.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image064.gif                   By symmetry, there is a second possibility at

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image066.gif

 

Since the point (-3, -4) is located in the 3rd quadrant, we can determine that the second angle is the one we need.  The polar coordinates of this point are File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image068.gif.

 

 

Try it Now

  1. Convert the following.

a. Convert polar coordinates File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image070.gif to File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image072.gif.

b. Convert Cartesian coordinates File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image074.gif to File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif.

Polar Equations

Just as a Cartesian equation like File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image077.gif describes a relationship between x and y values on a Cartesian grid, a polar equation can be written describing a relationship between r and θ values on the polar grid. 

 

 

Example 5

Sketch a graph of the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image079.gif.

 

The equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image079.gif describes all the points for which the radius r is equal to the angle.  To visualize this relationship, we can create a table of values.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image081.jpgText Box: θ	0	π/4	π/2	3π/4	π	5π/4	3π/2	7π/4	2π
r	0	π/4	π/2	3π/4	π	5π/4	3π/2	7π/4	2π

 

We can plot these points on the plane, and then sketch a curve that fits the points.  The resulting graph is a spiral.

 

Notice that the resulting graph cannot be the result of a function of the form y = f(x), as it does not pass the vertical line test, even though it resulted from a function giving r in terms of θ.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image084.jpg

Although it is nice to see polar equations on polar grids, it is more common for polar graphs to be graphed on the Cartesian coordinate system, and so, the remainder of the polar equations will be graphed accordingly. 

 

The spiral graph above on a Cartesian grid is shown here.

 

 

Example 6

Sketch a graph of the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image086.gif.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image088.jpg

Recall that when a variable does not show up in the equation, it is saying that it does not matter what value that variable has; the output for the equation will remain the same.

 

For example, the Cartesian equation y = 3 describes all the points where y = 3, no matter what the x values are, producing a horizontal line.

 

Likewise, this polar equation is describing all the points at a distance of 3 from the origin, no matter what the angle is, producing the graph of a circle.

The normal settings on graphing calculators and software graph on the Cartesian coordinate system with y being a function of x,  where the graphing utility asks for f(x), or simply y =.

 

To graph polar equations, you may need to change the mode of your calculator to Polar.  You will know you have been successful in changing the mode if you now have r as a function of θ, where the graphing utility asks for r(θ), or simply r =.

 

 

Example 7

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image090.jpgSketch a graph of the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image092.gif, and find an interval on which it completes one cycle.

 

While we could again create a table, plot the corresponding points, and connect the dots, we can also turn to technology to directly graph it.  Using technology, we produce the graph shown here, a circle passing through the origin.

 

Since this graph appears to close a loop and repeat itself, we might ask what interval of θ values yields the entire graph.  At θ = 0, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image094.gif.  We would then consider the next θ value when r will be 4, which would mean we are back where we started.  Solving,

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image096.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image098.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image100.gifor File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image102.gif

This shows us at 0 radians we are at the point (0, 4), and again atFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image104.gif radians we are at the point (0, 4) having finished one complete revolution.

 

The interval File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image106.gifyields one complete iteration of the circle.

 

 

Try it Now

3. Sketch a graph of the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image108.gif, and find an interval on which it completes one cycle.

 

 

The last few examples have all been circles.  Next we will consider two other “named” polar equations, limaçons and roses

 

 

Example 8

Sketch a graph of the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image110.gif.  What interval of θ values corresponds to the inner loop?

 

This type of graph is called a limaçon

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image112.jpgUsing technology, we can draw a graph.  The inner loop begins and ends at the origin, where r = 0.  We can solve for the θ values for which r = 0.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image114.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image116.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image118.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image120.gif or File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image122.gif

 

This tells us that r = 0, or the graph passes through the origin, twice on the interval

[0, 2π).

The inner loop arises from the interval File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image124.gif.  This corresponds to where the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image110.gif takes on negative values.

 

 

Example 9

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image126.jpgSketch a graph of the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image128.gif.  What interval of θ values describes one small loop of the graph?

 

This type of graph is called a 3 leaf rose.

 

Again we can use technology to produce a graph.  The interval [0, π) yields one cycle of this function.  As with the last problem, we can note that there is an interval on which one loop of this graph begins and ends at the origin, where r = 0.  Solving for θ,

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image130.gif                          Substitute u = 3θ

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image132.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image134.gif or File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image136.gif or File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image138.gif

Undo the substitution

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image140.gif         or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image142.gif         or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image144.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image146.gif           or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image148.gif             or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image150.gif

 

There are 3 solutions on File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image106.gif which correspond to the 3 times the graph returns to the origin, but the first two solutions we solved for above are enough to conclude that one loop corresponds to the interval File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image153.gif.   

Text Box: θ 	r	x	y
0	1	1	0
 
0	0	0
 
-1	 
 

 
0	0	0

If we wanted to get an idea of how the computer drew this graph, consider when θ = 0. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image161.gif, so the graph starts at (1,0).  As we found above, at File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image163.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image165.gif, the graph is at the origin.  Looking at the equation, notice that any angle in between File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image167.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image169.gif, for example at File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image038.gif, produces a negative r: File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image172.gif.  Notice that with a negative r value and an angle with terminal side in the first quadrant, the corresponding Cartesian point would be in the third quadrant.  Since File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image128.gif is negative on File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image153.gif, this interval corresponds to the loop of the graph in the third quadrant.

 

 

Try it Now

4. Sketch a graph of the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image174.gif.  Would you call this function a limaçon or a rose?

 

 

Converting Equations

While many polar equations cannot be expressed nicely in Cartesian form (and vice versa), it can be beneficial to convert between the two forms, when possible.  To do this we use the same relationships we used to convert points between coordinate systems.

 

 

Example 10

Rewrite the Cartesian equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image176.gif as a polar equation.

 

We wish to eliminate x and y from the equation and introduce r and θ.  Ideally, we would like to write the equation with r isolated, if possible, which represents r as a function of θ.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image176.gif                                    Remembering File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif we substitute

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image180.gif                                            File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image032.gif and so we substitute again

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image183.gif                                  Subtract File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image185.gif from both sides

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image187.gif                             Factor

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image189.gif                             Use the zero factor theorem

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image191.gif  or  r = 0                       Since r = 0 is only a point, we reject that solution.

 

The solution File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image191.gif is fairly similar to the one we graphed in Example 7.  In fact, this equation describes a circle with bottom at the origin and top at the point (0, 6).

 

Example 11

Rewrite the Cartesian equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image193.gif as a polar equation.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image193.gif                                        Use File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image032.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image028.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image197.gif                    Move all terms with r to one side

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image199.gif                    Factor out r

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image201.gif                    Divide

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image203.gif

 

In this case, the polar equation is more unwieldy than the Cartesian equation, but there are still times when this equation might be useful.

 

 

Example 12

Rewrite the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image205.gif as a Cartesian equation.

 

We want to eliminate θ and r and introduce x and y.  It is usually easiest to start by clearing the fraction and looking to substitute values that will eliminate θ.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image205.gif                                Clear the fraction

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image208.gif                            Use File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image026.gif to eliminate θ          

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image210.gif                                   Distribute and simplify

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image212.gif                                         Isolate the r

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image214.gif                                         Square both sides

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image216.gif                                   Use File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image218.gif

 

When our entire equation has been changed from r and θ to x and y we can stop unless asked to solve for y or simplify.

 

In this example, if desired, the right side of the equation could be expanded and the equation simplified further.  However, the equation cannot be written as a function in Cartesian form.

 

 

Try it Now

5. a. Rewrite the Cartesian equation in polar form: File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image220.gif

    b. Rewrite the polar equation in Cartesian form: File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image222.gif

Example 13

Rewrite the polar equation File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image224.gif in Cartesian form.

 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image224.gif                                       Use the double angle identity for sine

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image227.gif                           Use File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image026.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image030.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image230.gif                                      Simplify

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image232.gif                                            Multiply by r2

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image234.gif                                           Since File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image236.gif

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image238.gif

 

This equation could also be written as

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image240.gif   or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image242.gif

 

 

Important Topics of This Section

Cartesian coordinate system

Polar coordinate system

Plotting points in polar coordinates

Converting coordinates between systems

Polar equations: Spirals, circles, limaçons and roses

Converting equations between systems

 

 

Try it Now Answers

 

B

C

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image243.gifFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image244.gif

A

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image245.gif1. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image246.gif

 

2. a. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image070.gifconverts to  File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image249.gif

    b. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image251.gif converts to File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image253.gif

3. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image255.jpg  It completes one cycle on the interval File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image106.gif.

 

4. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image257.jpg  This is a 4-leaf rose.

 

5. a. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image220.gif becomes r = 3

    b. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image222.gif becomes File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image259.gif

 


Section 8.2 Exercises

 

Convert the given polar coordinates to Cartesian coordinates.

1. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image261.gif                  2. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image263.gif                  3. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image265.gif                  4. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image267.gif     

5. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image269.gif                  6. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image271.gif                7. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image273.gif                    8. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image275.gif         

9. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image277.gif                  10. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image279.gif              11. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image281.gif                     12. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image283.gif

 

Convert the given Cartesian coordinates to polar coordinates.

13. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image285.gif                     14. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image287.gif                     15. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image289.gif                   16. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image291.gif      

17. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image293.gif                   18. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image295.gif                   19. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image297.gif                        20. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image299.gif

 

Convert the given Cartesian equation to a polar equation.

21. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image301.gif                     22. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image303.gif                    23. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image305.gif                 24. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image307.gif

25. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image309.gif         26. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image311.gif          27. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image313.gif           28. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image315.gif

 

Convert the given polar equation to a Cartesian equation.

29. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image317.gif                                              30. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image319.gif        

31. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image321.gif                               32. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image323.gif

33. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image325.gif                                              34. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image327.gif                     

35. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image329.gif                                     36. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image331.gif

 

 

 

 

 

 

Match each equation with one of the graphs shown.

37. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image333.gif                38. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image335.gif                39. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image337.gif   

40. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image339.gif                41. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image341.gif                                 42. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image343.gif

Ahttp://www.wamap.org/filter/graph/svgimg.php?sscr=-1%2C5%2C-3%2C3%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C2%2B2cos%28t%29%2Cnull%2C0%2C0%2C0%2C6.4%2Cblack%2C1%2Cnone       Bhttp://www.wamap.org/filter/graph/svgimg.php?sscr=-2%2C8%2C-5%2C5%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C3%2B4cos%28t%29%2Cnull%2C0%2C0%2C0%2C6.4%2Cblack%2C1%2Cnone       Chttp://www.wamap.org/filter/graph/svgimg.php?sscr=-2%2C8%2C-5%2C5%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C4%2B3cos%28t%29%2Cnull%2C0%2C0%2C0%2C6.4%2Cblack%2C1%2Cnone

Dhttp://www.wamap.org/filter/graph/svgimg.php?sscr=-3%2C3%2C-1%2C5%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C2%2B2sin%28t%29%2Cnull%2C0%2C0%2C0%2C6.4%2Cblack%2C1%2Cnone       Ehttp://www.wamap.org/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C4%2Cnull%2C0%2C0%2C0%2C6.4%2Cblack%2C1%2Cnone       Fhttp://www.wamap.org/filter/graph/svgimg.php?sscr=-1.5%2C1.5%2C-.5%2C2.5%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C2sin%28t%29%2Cnull%2C0%2C0%2C0%2C6.4%2Cblack%2C1%2Cnone

 

Match each equation with one of the graphs shown. 

43. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image357.gif                        44. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image359.gif                     45. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image361.gif         

46. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image363.gif           47. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image365.gif   48. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image367.gif

Ahttp://www.wamap.org/filter/graph/svgimg.php?sscr=-4%2C4%2C-4%2C4%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2Ct*cos%28t%29%2Cnull%2C0%2C0%2C-6.28%2C6.28%2Cblack%2C1%2Cnone         Bhttp://www.wamap.org/filter/graph/svgimg.php?sscr=-2%2C2%2C-2%2C2%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C1%2Bsin%282*t%29%2Cnull%2C0%2C0%2C0%2C6.28%2Cblack%2C1%2Cnone         C http://www.wamap.org/filter/graph/svgimg.php?sscr=-2%2C2%2C-2%2C2%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2Clog%28t%29%2Cnull%2C0%2C0%2C0.001%2C20%2Cblack%2C1%2Cnone

Dhttp://www.wamap.org/filter/graph/svgimg.php?sscr=-6%2C6%2C-6%2C6%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C5cos%28t%2F2%29%2Cnull%2C0%2C0%2C-6.28%2C6.28%2Cblack%2C1%2Cnone         Ehttp://www.wamap.org/filter/graph/svgimg.php?sscr=-3%2C3%2C-2%2C4%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C8*sin%28t%29*cos%28t%29*cos%28t%29%2Cnull%2C0%2C0%2C0%2C6.28%2Cblack%2C1%2Cnone          Fhttp://www.wamap.org/filter/graph/svgimg.php?sscr=-3%2C3%2C-4%2C2%2C7%2C7%2Cnull%2Cnull%2Cnull%2C300%2C300%2Cpolar%2C1%2B2sin%283*t%29%2Cnull%2C0%2C0%2C0%2C6.28%2Cblack%2C1%2Cnone

 

 

 

 

Sketch a graph of the polar equation.

49. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image381.gif                      50. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image383.gif                      51. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image385.gif       

52. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image387.gif                    53. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image389.gif                    54. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image391.gif                   

55. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image393.gif                    56. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image395.gif                   57. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image333.gif   

58. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image398.gif                 59. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image400.gif                 60. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image402.gif

61. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image404.gif       ­­­                        62. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image406.gif                               

63. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image408.gif, a conchoid                        64. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image410.gif, a lituus[1]

65. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image412.gif, a cissoid                  



[1] This curve was the inspiration for the artwork featured on the cover of this book.

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)