Answer
(a) (i) The polar coordinates are $(4\sqrt{2}, \frac{3\pi}{4})$
(ii) The polar coordinates are $(-4\sqrt{2}, \frac{7\pi}{4})$
(b) (i) The polar coordinates are $(6, \frac{\pi}{3})$
(ii) The polar coordinates are $(-6, \frac{4\pi}{3})$
Work Step by Step
(a) The Cartesian coordinates are $(-4,4)$
(i) We can find the distance from the origin:
$r = \sqrt{(-4)^2+(4)^2} = 4\sqrt{2}$
The angle is $~~\pi - tan^{-1}(\frac{4}{4}) = \frac{3\pi}{4}$
The polar coordinates are $(4\sqrt{2}, \frac{3\pi}{4})$
(ii) The distance from the origin is $4\sqrt{2}$
Then $r = -4\sqrt{2}$
The angle is $\frac{3\pi}{4}+\pi = \frac{7\pi}{4}$
The polar coordinates are $(-4\sqrt{2}, \frac{7\pi}{4})$
(b) The Cartesian coordinates are $(3,3\sqrt{3})$
(i) We can find the distance from the origin:
$r = \sqrt{(3)^2+(3\sqrt{3})^2} = 6$
The angle is $tan^{-1} (\frac{3\sqrt{3}}{3}) = \frac{\pi}{3}$
The polar coordinates are $(6, \frac{\pi}{3})$
(ii) The distance from the origin is $6$
Then $r = -6$
The angle is $\frac{\pi}{3}+\pi = \frac{4\pi}{3}$
The polar coordinates are $(-6, \frac{4\pi}{3})$
