# How do you graph r^2 = sin(2theta)?

Sep 11, 2016

#### Explanation:

Try plotting with points.

Some easy points would be $\left(0 , 0\right)$, $\left(\frac{\pi}{12} , \pm \frac{1}{\sqrt{2}}\right)$, $\left(\frac{\pi}{8} , \pm \frac{1}{\sqrt[4]{2}}\right)$, $\left(\frac{\pi}{6} , \pm \frac{\sqrt[4]{3}}{\sqrt{2}}\right)$, $\left(\frac{\pi}{4} , \pm 1\right)$, $\left(\frac{3 \pi}{8} , \pm \frac{1}{\sqrt[4]{2}}\right)$, $\left(\frac{\pi}{2} , 0\right)$.

From the equation, $\sin \left(2 \theta\right)$ cannot be negative, so $\theta$ is restricted to $0 \le \theta \le \frac{\pi}{2}$ or $\pi \le \theta \le \frac{3 \pi}{2}$.

And most values of $\theta$ corresponds to a positive and a negative value of $r$, the graph should have rotational symmetry.

If a graphing program for cartesian is available, the cartesian equation is

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