# How do you graph the system of polar equations to solve r=2sintheta and r=2sin2theta?

Jul 3, 2018

$\left(r , \theta\right) = \left(0 , 0\right)$ and $\left(\sqrt{3} , \frac{\pi}{3}\right)$.

#### Explanation:

$r = 2 \sin \theta = 2 \sin 2 \theta \ge 0$ gives

$\sin \theta - 2 \sin \theta \cos \theta = \sin \theta \left(1 - 2 \cos \theta\right) = 0$.

So, the factor $\sin \theta = 0$ gives

$\theta = k \pi , k = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$.

Corresponding r = 0.

The factor $2 \cos \theta - 1 = 0$ sets

$\cos \theta = \frac{1}{2} = \cos \left(\frac{\pi}{3}\right)$, and this gives

$\theta = 2 n \pi \pm \frac{\pi}{3} , n = 0 , \pm 1 , \pm 2 , \pm 3 , \ldots$

Here, only for $\theta = 2 n \pi + \frac{\pi}{3}$,

$r = 2 \sin \theta = 2 \sin 2 \theta = \sqrt{3} > 0$.

Thus, combining both the sets, the common points are

$\left(r , \theta\right) = \left(0 , 0\right)$ and $\left(\sqrt{3} , \frac{\pi}{3}\right)$. See graph.

Use $\left(x , y\right) = r \left(\cos \theta , \sin \theta\right)$.

graph{(x^2 + y^2 - 2y)((x^2 + y^2)^1.5-4xy)(x^2+y^2-.005)( (x-0.866)^2+(y-1.5)^2-.005)=0[-4 4 -2 2]}