Using a Table of Values to Sketch Polar Coordinate Curves

If we don't have a computer and we need to sketch the function on paper, we need to set up a table of values as follows:

θ (degrees) `0°` `30°` `60°` `90°` `120°` `150°` `180°`
θ (radians) `0`` `π/6` `π/3` `π/2` `(2π)/3` `(5π)/6` `π`

r = 2 + 3 sin θ

`2` `3.5` `4.60` `5` `4.6` `3.5` `2`

θ (degrees) `180°` `210°` `240°` `270°` `300°` `330°` `360°`
θ (radians) `π` `(7π)/6` `(4π)/3` `(3π)/2` `(5π)/3` `(11π)/6` `2π`

r = 2 + 3 sin θ

`2` `0.5` `-0.60` `-1` `-0.60` `0.5` `2`

 

The first 7 points from this table are (2, 0°), (3.5, 30°), (4.60, 60°), (5, 90°), (4.6, 120°), (3.5, 150°), and (2, 180°).

We plot these points (they are numbered) on the polar graph. I have also indicated with arrows the direction you need to go when joining the points.

limacon from table of values

Recall: A negative "r" means we need to be on the opposite side of the origin.

Here's the complete graph.

30°60°90°120°150°180°210°240°270°300°330°012345

Graph of r = 2 + 3 sin θ, a limacon.

Conversion to Rectangular Form

Once again we convert our polar function into rectangular form so we can see how much simpler polar form is for certain functions.

`r = 2 + 3\ sin\ θ`

In rectangular form, this is:

`sqrt(x^2+y^2)=2+3y/sqrt(x^2+y^2)`

`x^2+y^2=2sqrt(x^2+y^2)+3y`

`x^2+y^2-3y=2sqrt(x^2+y^2)`

`(x^2+y^2-3y)^2=4(x^2+y^2)`

`x^4+2x^2y^2+y^4-6y(x^2+y^2)+` `9y^2` `=4(x^2+y^2)`

`x^4+2x^2y^2+y^4-6y(x^2+y^2)-` `4x^2+` `5y^2` `=0`

Notice how much simpler the polar form is compared to the rectangular form.