Angle of intersection of two polar curves

Question

I'm trying to find the angle of intersection between two polar curves:

$$\begin{cases}r= 5 + 3 \sin\theta \\ r' = 3\end{cases}$$

I've set them as $5 + 3\sin\theta = 3$ and got to $\sin\theta = -2/3$ but from here how do I find $\theta$? The unit circle seems useless here. Can someone please teach me the way?

Thanks

Answer 1

$$\sin \theta = -\frac 23 \iff \arcsin (\sin \theta) = \arcsin\left(-\frac 23\right) \iff \theta =\arcsin\left(-\frac 23\right)$$

You'll need to approximate $\theta$, since there is no "nice" angle $\theta$ such that $\sin(\theta)=-\frac 23$.

Answer 2

This is really just a footnote to amWhy's answer. If you graph the two equations in your system you'll get something like:

$\phantom{XXXXXXXXX}$

So there are two points where the two curves meet. The angles at which they meet is given by: $$\arcsin\left(-\dfrac23\right)\quad\color{grey}{\text{ and }}\quad\pi-\arcsin\left(-\dfrac23\right).$$