Originally Posted by

**Jukodan** I recently came across this problem and would like some clarification as to why one of the answers to theta cannot be $\displaystyle \frac{\pi }{2}$

The problem asks to sketch two polar curves and find where they intersect, and then present those points in rectangular coordinates. I can easily see from the graph that the points are (1,0) and (-1,0). The two polar functions are,

$\displaystyle r= \cos ^{2}\theta $ and r = -1.

I set them equal to each other

$\displaystyle \cos ^{2}\theta= 1$

Ok, is it r = 1 or r = -1??

$\displaystyle \frac{1}{2}\left ( 1+\cos 2\theta \right )= 1$

$\displaystyle 1+\cos 2\theta = 2$

$\displaystyle \cos 2\theta = 1$

So theta should be...$\displaystyle 0,\pi,-\pi,\frac{-\pi}{2},\frac{\pi}{2}$

This is wrong: $\displaystyle \cos 2\theta=1\Longrightarrow 2\theta=2k\pi\,,\,\,k\in\mathbb{Z} \Longrightarrow \theta = k\pi = 0,\,\pm \pi,\,\pm 2\pi,\ldots$

Yet the graph tells me only pi and -pi can be the answer. I'm confused...