Answer
$a.\quad(1,1)$
$b.\quad (1,0)$
$c.\quad (0,0)$
$d.\quad (-1,-1)$
$e.\displaystyle \quad (\frac{3\sqrt{3}}{2},-\frac{3}{2})$
$f.\quad (3,4)$
$g.\quad (1,0)$
$h.\quad (-\sqrt{3},3)$
Work Step by Step
$ a.\quad$
Polar: $(r,\theta)=(\sqrt{2},\pi/4)$
Cartesian: $(r\displaystyle \cos\theta,r\sin\theta)=(\sqrt{2}\cdot\frac{1}{\sqrt{2}},\sqrt{2}\cdot\frac{1}{\sqrt{2}})=(1,1)$
$ b.\quad$
Polar: $(r,\theta)=(1,0)$
Cartesian: $(r\cos\theta,r\sin\theta)=(1\cdot 1,1\cdot 0)=(1,0)$
$ c.\quad$
Polar: $(r,\theta)=(0,\pi/2)$
Cartesian: $(r\cos\theta,r\sin\theta)=(0,0)$
$ d.\quad$
Polar: $(r,\theta)=(-\sqrt{2},\pi/4)$
Cartesian: $(r\displaystyle \cos\theta,r\sin\theta)=(-\sqrt{2}\cdot\frac{1}{\sqrt{2}},-\sqrt{2}\cdot\frac{1}{\sqrt{2}})=(-1,-1)$
$ e.\quad$
Polar: $(r,\theta)=(-3,5\pi/6)$
Cartesian: $(r\displaystyle \cos\theta,r\sin\theta)=(-3\cdot(-\frac{\sqrt{3}}{2}),-3\cdot\frac{1}{2})=(\frac{3\sqrt{3}}{2},-\frac{3}{2})$
$ f.\quad$
Polar: $(r,\displaystyle \theta)=(5,\tan^{-1}\frac{4}{3})$
Cartesian: $(r\cos\theta,r\sin\theta)=(5\cdot 0.6,5\cdot 0.8)=(3,4)$
$ g.\quad$
Polar: $(r,\theta)=(-1,7\pi)$
Cartesian: $(r\cos\theta,r\sin\theta)=(-1\cdot(-1),-1\cdot 0)=(1,0)$
$ h.\quad$
Polar: $(r,\theta)=(2\sqrt{3},2\pi/3)$
Cartesian: $(r\displaystyle \cos\theta,r\sin\theta)=(2\sqrt{3}\cdot(-\frac{1}{2}),2\sqrt{3}\cdot(\frac{\sqrt{3}}{2}))=(-\sqrt{3},3)$
