find all points of intersection between the curves. (r = sqrt{3}cos(3\theta)) and (r=sin(3\theta)). enter…

Explanation:

Step1: Set the two equations equal

Set $\sqrt{3}\cos(3\theta)=\sin(3\theta)$. Then $\frac{\sin(3\theta)}{\cos(3\theta)}=\sqrt{3}$, so $\tan(3\theta)=\sqrt{3}$.

Step2: Solve for $3\theta$

We know that $\tan(3\theta)=\sqrt{3}$, so $3\theta = n\pi+\frac{\pi}{3}$, where $n\in\mathbb{Z}$. Then $\theta=\frac{n\pi}{3}+\frac{\pi}{9}$.

Step3: Find values of $\theta$ in the range $[0, 2\pi)$

When $n = 0$, $\theta=\frac{\pi}{9}$; when $n = 1$, $\theta=\frac{\pi}{3}+\frac{\pi}{9}=\frac{4\pi}{9}$; when $n = 2$, $\theta=\frac{2\pi}{3}+\frac{\pi}{9}=\frac{7\pi}{9}$; when $n = 3$, $\theta=\pi+\frac{\pi}{9}=\frac{10\pi}{9}$; when $n = 4$, $\theta=\frac{4\pi}{3}+\frac{\pi}{9}=\frac{13\pi}{9}$; when $n = 5$, $\theta=\frac{5\pi}{3}+\frac{\pi}{9}=\frac{16\pi}{9}$.

Step4: Find the $r$ - values

Substitute $\theta$ into either $r = \sin(3\theta)$ or $r=\sqrt{3}\cos(3\theta)$. For $\theta=\frac{\pi}{9}$, $r=\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$; for $\theta=\frac{4\pi}{9}$, $r=\sin(\frac{4\pi}{3})=-\frac{\sqrt{3}}{2}$; for $\theta=\frac{7\pi}{9}$, $r=\sin(\frac{7\pi}{3})=\frac{\sqrt{3}}{2}$; for $\theta=\frac{10\pi}{9}$, $r=\sin(\frac{10\pi}{3})=-\frac{\sqrt{3}}{2}$; for $\theta=\frac{13\pi}{9}$, $r=\sin(\frac{13\pi}{3})=\frac{\sqrt{3}}{2}$; for $\theta=\frac{16\pi}{9}$, $r=\sin(\frac{16\pi}{3})=-\frac{\sqrt{3}}{2}$. In polar coordinates, the points of intersection are $(\frac{\sqrt{3}}{2},\frac{\pi}{9}),(-\frac{\sqrt{3}}{2},\frac{4\pi}{9}),(\frac{\sqrt{3}}{2},\frac{7\pi}{9}),(-\frac{\sqrt{3}}{2},\frac{10\pi}{9}),(\frac{\sqrt{3}}{2},\frac{13\pi}{9}),(-\frac{\sqrt{3}}{2},\frac{16\pi}{9})$. We can also consider the case when $r = 0$. Set $r=\sqrt{3}\cos(3\theta)=0$, then $3\theta=(2k + 1)\frac{\pi}{2}$, $\theta=(2k + 1)\frac{\pi}{6}$. Set $r=\sin(3\theta)=0$, then $3\theta = m\pi$, $\theta=\frac{m\pi}{3}$. The common - zero cases occur when $\theta = 0,\frac{\pi}{3},\frac{2\pi}{3},\pi,\frac{4\pi}{3},\frac{5\pi}{3}$. When $\theta = 0$, $r = 0$; when $\theta=\frac{\pi}{3}$, $r = 0$; when $\theta=\frac{2\pi}{3}$, $r = 0$; when $\theta=\pi$, $r = 0$; when $\theta=\frac{4\pi}{3}$, $r = 0$; when $\theta=\frac{5\pi}{3}$, $r = 0$.

Answer:

$(0,0),(\frac{\sqrt{3}}{2},\frac{\pi}{9}),(-\frac{\sqrt{3}}{2},\frac{4\pi}{9}),(\frac{\sqrt{3}}{2},\frac{7\pi}{9}),(-\frac{\sqrt{3}}{2},\frac{10\pi}{9}),(\frac{\sqrt{3}}{2},\frac{13\pi}{9}),(-\frac{\sqrt{3}}{2},\frac{16\pi}{9})$