[Math] Angle of intersection of two polar curves

polar coordinates

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

Best Answer

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

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

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