I am learning pre-calculus and I am not able to answer this question from the textbook:
Find the rectangular coordinates of all the points of intersection of the two polar curves ${\sqrt 3}\sin\theta=r$ and $\cos\theta=r$
Intersection coordinates of the two polar curves
algebra-precalculuspolar coordinates
Related Solutions
I just added this plot to show what is Raskolnikov trying to tell you. You can find the area. Note the integral he wrote above.
$$\sin \theta = -\frac 23 \iff \arcsin (\sin \theta) = \arcsin\left(-\frac 23\right) \iff \theta =\arcsin\left(-\frac 23\right)$$
You'll need to approximate $\theta$, since there is no "nice" angle $\theta$ such that $\sin(\theta)=-\frac 23$.
Best Answer
Dealing with intersections of polar curves can be tricky. It would be awesome if we could just set the two equations equal to each other and solve for $\theta$. $$\sqrt3\sin\theta=\cos\theta\\ \tan\theta=\frac1{\sqrt 3}\\ \theta=\frac\pi6,\frac{7\pi}6\\ (r,\theta)=\left(\frac{\sqrt3}2,\frac\pi6\right),\left(\frac{-\sqrt3}2,\frac{7\pi}6\right)\\ (x,y)=\left(\frac34,\frac{\sqrt3}{4}\right)$$
because "both" of those polar coordinates are the same point (RED FLAG $1$). But let's double-check our work by graphing both of those polar curves in Desmos:
The curves also meet at the origin. Why didn't we catch that? Because the first curve goes through the origin at $(0,0)$ and $(0,\pi)$ and the second curve goes through the origin at $(0,\frac\pi2)$ and $(0,\frac{3\pi}2)$, and those are all the same point in polar coordinates (RED FLAG $2$).
The moral of the story is to always graph your polar curves to make sure that you're catching all of the intersection points.