Answer
(a) (i) The polar coordinates are $(2, \frac{11\pi}{6})$
(ii) The polar coordinates are $(-2, \frac{5\pi}{6})$
(b) (i) The polar coordinates are $(6, \pi)$
(ii) The polar coordinates are $(-6, 0)$
Work Step by Step
(a) The Cartesian coordinates are $(\sqrt{3},-1)$
(i) We can find the distance from the origin:
$r = \sqrt{(\sqrt{3})^2+(-1)^2} = 2$
The angle is $~~tan^{-1}(\frac{-1}{\sqrt{3}})+2\pi = \frac{11\pi}{6}$
The polar coordinates are $(2, \frac{11\pi}{6})$
(ii) The distance from the origin is $2$
Then $r = -2$
The angle is $\frac{11\pi}{6}-\pi = \frac{5\pi}{6}$
The polar coordinates are $(-2, \frac{5\pi}{6})$
(b) The Cartesian coordinates are $(-6,0)$
(i) We can find the distance from the origin:
$r = \sqrt{(-6)^2+(0)^2} = 6$
The angle is $\pi$
The polar coordinates are $(6, \pi)$
(ii) The distance from the origin is $6$
Then $r = -6$
The angle is $\pi-\pi = 0$
The polar coordinates are $(-6, 0)$
