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.
Recall: A negative "r" means we need to be on the opposite side of the origin.
Here's the complete graph.
Graph of r = 2 + 3 sin θ, a limacon.
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.