Finding points of intersection of two curves

Question

I have a problem where I needed to find the intersection of two curves, with $r=3+\sin(\theta)$ and $r=7\sin(\theta)$ for $[0<\theta<2\pi]$

I have that it's supposed to be $(7/6,\pi/6)$ and $(7/6,5\pi/6)$. The way I tried to do the problem based on the few examples my book gave is:

$3+\sin(\theta)=7\sin(\theta)$

$3=6\sin(\theta)$

$1=3\sin(\theta)$

$\sin(\theta)=1/3$

But I can't see any way to get the right values from that. What am I missing here? Am I missing something simple, completely misunderstanding how the problem is done?

Answer 1

Use the inverse function $\arcsin$ to get a value of $\theta$ in $(0,\pi/2)$. Then have a closer look at the $\sin$ function to get the other solution in $(\pi/2,\pi)$.

Answer 2

I plotted your curves and your solutions and they seem to match:

(Large Version)