We need to sketch `r = 2 + 3 sin θ`.

Let's use the following interactive graph to understand what is going on. You will trace out the required curve as you change the angle.

Drag the blue dot (slowly) left and right to change the angle θ and observe the resulting changes in the value of r.

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Hint: Let go of the blue dot to see a smooth curve at any time. You can trace out positive or negative angles.

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.