I'm looking at some notes that I was given for my Calculus II class on converting from Cartesian to polar coordinates. Now I understand how to solve for r and $\theta $ but I'm looking at how she solved these two and maybe some can clear the air on these two problems.
Qaere: Cartesian coordinates are given. Write in polar coordinates with $0 \le \theta \le 2\pi$ and (i) $r>0$ and (ii) $r<0$.
(a) $\bigl(-1, \sqrt{3}\bigr)$
So, $(-1)^2+\bigl(-\sqrt{3}\bigr)^2=r^2$ gets us $r=\pm 2$.
(i) $\bigl(2,\frac{4\pi}3\bigr)$
(ii) $\bigl(-2, \frac \pi 3\bigr)$
So actually as I'm writing this (a) makes sense.
Now for (b) $(-2,3)$
$r=\sqrt{13}$
(i) $\left(\sqrt{13},\arctan\left(-\frac 3 2\right)+\pi\right)$
(ii) $\left(-\sqrt{13}, \arctan\left(-\frac 3 2\right)+2\pi\right)$
Now that I look at the values for each that end of in the corresponding quadrants. But maybe someone can just give me a better way to do these problem so I'm not wasting time graphing them out. Or maybe that's the best way to go about them to reduce any error. Thanks!
Some points for the first one (a). As you have achieved correctly $\theta=\frac{-\pi}3$ and $r=\pm 2$. Now consider the following graphs: