Answer
(a) $(5,90^\circ),(5,-270^\circ)$
(b) $(2\sqrt 2,225^\circ),(2\sqrt 2,-135^\circ)$
Work Step by Step
(a) Given the coordinates $(0,5)$ on the positive y-axis, we have $x=r\ cos\theta =0, y=r\ sin\theta = 5$, thus $r=\sqrt {0^2+5^2}=5$ and $cot\theta=0, \theta=90^\circ, -270^\circ ...$. Two pairs of polar coordinates $(5,90^\circ),(5,-270^\circ)$
(b) Given the coordinates $(-2,-2)$ in quadrant III, we have $x=r\ cos\theta =-2, y=r\ sin\theta = -2$, thus $r=\sqrt {(-2)^2+(-2)^2}=2\sqrt 2$ and $tan\theta=1, \theta=225^\circ, -135^\circ ...$. Two pairs of polar coordinates $(2\sqrt 2,225^\circ),(2\sqrt 2,-135^\circ)$
