Question

find all points of intersection of the graphs r=1+sin(theta) and r=3sin(theta). Write your points in polar coordinates. I found two points so far : (3/2,pi/5) and (3/2, 5pi/6). I know it intersects at the pole at theta= 4.712 but how do you find that?

Answer 1

\[\begin{align*}
1+\sin\theta&=3\sin\theta\\\\
1&=2\sin\theta\\\\
\frac{1}{2}&=\sin\theta\\\\
\theta&=\dfrac{\pi}{6},~\dfrac{5\pi}{6}\quad\text{(assuming angles in one rev. of the unit circle)}\end{align*}\]
The "solution" at \(\dfrac{3\pi}{2}\) doesn't exist, although it is technically an intersection point. In particular, there is no value of \(\theta\) for which both curves pass through the origin.

For the circle (\(3\sin\theta\)), the curve passes through the origin for \(\theta=0,\pi,2\pi\).

In contrast, the cardioid (\(1+\sin\theta\)) passes through it for \(\theta=\dfrac{3\pi}{2}\).

Answer 2

Thank you :)