Intersection coordinates of the two polar curves

Question

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 $\cos\theta=r$

Answer 1

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.

Answer 2

Use that $$\tan(\theta)=\frac{1}{\sqrt{3}}$$

Answer 3

Hint: If the two of them intersect, then there is some $r,\theta$ satisfying the two equations simultaneously.