# How do you graph r^2=3sin2θ?

Dec 12, 2016

#### Explanation:

The relation between polar coordinates $\left(r , \theta\right)$ and Cartesian coordinates $\left(x , y\right)$ is $x = r \cos \theta$, $y = r \sin \theta$ and ${r}^{2} = {x}^{2} + {y}^{2}$.

We can use this to convert equation in polar coordinates to an equation with Cartesian coordinates.

${r}^{2} = 3 \sin 2 \theta$

$\Leftrightarrow {r}^{2} = 3 \times 2 \sin \theta \cos \theta$

or ${r}^{2} \times {r}^{2} = 6 \times r \sin \theta \times r \cos \theta$

or ${\left({x}^{2} + {y}^{2}\right)}^{2} = 6 x y$

Note that

(a) As $6 x y$ is a complete square, it is positive and hence curve can lie only in first and third quadrant.

(b) Further as maximum value of ${r}^{2} = 3 \sin 2 \theta$, maximum possible value for ${r}^{2}$ is $3$ and so $r$ cannot be more than $\sqrt{3} = 1.732 \ldots .$

(c) As replacing $x$ and $y$ with each other does not change the equation, it is symmetric along $x = y$.

Now we can put different values of $x$ to get $y$ (both less than $\sqrt{3}$) and draw the graph.

The function appears as follows.
graph{((x^2+y^2)^2-6xy)(x-y)=0 [-5, 5, -2.5, 2.5]}