How to sketch polar curves

 
 
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Identifying different forms of polar curves

We’ll sketch polar curves by plotting values for ???r??? at known values of ???\theta???. We can also use the table below to quickly graph polar curves given in these standard forms.

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Summary table of lines, circles, cardioids, limacons, etc.
 

If we can’t use the table above to find a standard form for the polar curve we’re given, then we can always generate a table of coordinate points ???(r,\theta)???. In order to do that, we’ll take the value inside the trigonometric function that includes ???\theta???, set it equal to ???\pi/2???, then solve for ???\theta???. For example, given the polar curve ???r=6\sin{3\theta}???,

???3\theta=\frac{\pi}{2}???

???\theta=\frac{\pi}{6}???

Then we’ll find ???r??? for the increments of ???\pi/6??? on the interval ???0\leq\theta\leq 2\pi???.

 
Table of r for increments of pi
 

Plotting these points on polar axes, we get

 
Sketch of the polar curve
 
 
 

Step-by-step example of how to sketch a polar curve


 
 
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Graphing lines in polar coordinates

Example

Graph the polar curves on the same axes.

???\theta=\frac{\pi}{3}???

???r\cos{\theta}=3???

???r\sin{\theta}=-2???


Using the table of standard curves, we can plot all of these on the same axes.

  1. ???\theta=\pi/3??? is like ???\theta=\beta???, so it’s a straight line through the origin at the angle ???\pi/3???.

  2. ???r\cos{\theta}=3??? is like ???r\cos{\theta}=a???, so it’s a vertical line through ???x=3???.

  3. ???r\sin{\theta}=-2??? is like ???r\sin{\theta}=b???, so it’s a horizontal line through ???y=-2???.

Graph of lines in polar coordinates

Let’s try some examples with circles defined in terms of polar coordinates.

Sketching polar curves for Calculus 2.jpg

To plot a polar curve, find points at increments of theta, then plot them on polar axes.

Graphing curves on the same set of axes

Example

Graph the polar curves on the same axes.

???r=4???

???r=6\cos{\theta}???

???r=-4\sin{\theta}???

???r=2\cos{\theta}+6\sin{\theta}???


Using the table of standard curves, we can plot all of these on the same axes.

  1. ???r=4??? is like ???r=a???, so it’s a circle centered at ???(0,0)??? with radius ???4???.

  2. ???r=6\cos{\theta}??? is like ???r=2a\cos{\theta}???, so it’s a circle centered at ???(3,0)??? with radius ???|3|???.

  3. ???r=-4\sin{\theta}??? is like ???r=2b\sin{\theta}???, so it’s a circle centered at ???(0,-2)??? with radius ???|-2|???.

  4. ???r=2\cos{\theta}+6\sin{\theta}??? is like ???r=2a\cos{\theta}+2b\sin{\theta}???, so it’s a circle centered at ???(1,3)??? with radius ???\sqrt{a^2+b^2}=\sqrt{10}???.

Sketch of polar curves on the same set of axes

Graphing cardioids, limacons, and roses

Example

Graph the polar curves.

???r=3+3\sin{\theta}???

???r=2+4\cos{\theta}???

???r=7+6\cos{\theta}???

???r=6\sin{2\theta}???


For ???r=3+3\sin{\theta}???:

???r=3+3\sin{\theta}??? is like ???r=a\pm{a}\sin{\theta}???, so it’s a cardioid through the origin. We’ll generate a table of values over the interval ???0\le\theta\le2\pi???.

Table of points for a cardioid

With these points and knowing the shape of our polar curve, we can sketch the graph.

Graph of a cardioid

For ???r=2+4\cos{\theta}???:

???r=2+4\cos{\theta}??? is like ???r=a\pm{b}\cos{\theta}??? with ???a<b???, so it’s a limaçon with an inner loop. We’ll generate a table of values over the interval ???0\le\theta\le2\pi???.

Table of points for a limacon

With these points and knowing the shape of our polar curve, we can sketch the graph.

Graph of a limacon

For ???r=7+6\cos{\theta}???:

???r=7+6\cos{\theta}??? is like ???r=a\pm{b}\cos{\theta}??? with ???a>b???, so it’s a limaçon without an inner loop. We’ll generate a table of values over the interval ???0\le\theta\le2\pi???.

Table of points for a limacon without an inner loop

With these points and knowing the shape of our polar curve we can sketch the graph.

Graph of a limacon without an inner loop

For ???r=6\sin{2\theta}???:

???r=6\sin{2\theta}??? doesn’t match any of the standard forms in our table. In this case, we’ll set the value inside our trigonometric function equal to ???\pi/2??? and then solve for ???\theta???.

???2\theta=\frac{\pi}{2}???

???\theta=\frac{\pi}{4}???

Then we’ll find ???r??? for the increments of ???\pi/4??? on the interval ???0\leq\theta\leq 2\pi???.

Table of points for a rose

Plotting these points on polar axes, we get

Graph of a rose
 
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