# What are all the intersections of the graphs r=sin2theta and r=sin3theta?

## What are all of the intersections of the graphs $r = \sin 2 \theta \mathmr{and} r = \sin 3 \theta$ ?

Apr 28, 2018

Solution for $\theta$ are given by $\theta = \left(2 n + 1\right) \frac{\pi}{5}$ or $\theta = 2 n \pi$, where $n$ is an integer. Polar coordinates of solution are {(0,0),(0.951,pi/5),(-0.588,(3pi)/5),(0,pi),(0.588,(7pi)/5),),(-0.951,(9pi)/5),(0,2pi)}

#### Explanation:

Intersection of graphs of $r = \sin 2 \theta$ and $r = \sin 3 \theta$ will be there where

$\sin 2 \theta = \sin 3 \theta$

or $\sin 3 \theta - \sin 2 \theta = 0$

or $2 \cos \left(\frac{3 \theta + 2 \theta}{2}\right) \sin \left(\frac{3 \theta - 2 \theta}{2}\right) = 0$

or $2 \cos \left(\frac{5 \theta}{2}\right) \sin \left(\frac{\theta}{2}\right) = 0$

i.e. either $\frac{5 \theta}{2} = \left(2 n + 1\right) \frac{\pi}{2}$ i.e. $\theta = \left(2 n + 1\right) \frac{\pi}{5}$

or $\frac{\theta}{2} = n \pi$ i.e. $\theta = 2 n \pi$, where $n$ is an integer.

In the interval $\left[{0}^{\circ} , {360}^{\circ}\right]$, solutions are $\left\{{0}^{\circ} , {36}^{\circ} , {108}^{\circ} , {180}^{\circ} , {252}^{\circ} , {324}^{\circ} , {360}^{\circ}\right\}$. In radians they are $\left\{0 , \frac{\pi}{5} , \frac{3 \pi}{5} , \pi , \frac{7 \pi}{5} , \frac{9 \pi}{5} , 2 \pi\right\}$

and putting this in $r = \sin 2 \theta$ or $r = \sin 3 \theta$, corresponding values of $r$ are $\left[0 , 0.951 , - 0.588 , 0 , 0.588 , - 0.951 , 0\right\}$

and polar coordinates of solution are {(0,0),(0.951,pi/5),(-0.588,(3pi)/5),(0,pi),(0.588,(7pi)/5),),(-0.951,(9pi)/5),(0,2pi)}