Sketching Complex region 

This is the question: 
Conisder the points in the region $R$ shown in the Argand diagram of Figure 2, consisting of all points in a right-angled sector of radius $1$, except for the point $ z =0.8 cis (\frac{\pi}{6}) $
Sktech the region containing all points $w^3$, where $w$ is a point within the region $R$




What I have tried:
The only thing I've gotten is that the eye should be located at $z^3$
If $z = rcis(x)$
$z =0.8 cis (\frac{\pi}{6}) $
$z^3 = 0.512i$ which is where the eye should be located
However I am having trouble drawing the rest of the diagram?\
How would I go about doing this...
 A: Hint: For the moment, forget about the point $z$. The filled-in region is the union of the collection of line segments $[0,\operatorname{cis} t]$ for $\frac{\pi}{12}\leq t\leq \frac{7\pi}{12}$.
The image of the radial segment $[0,\operatorname{cis} t]$ under the map $w\mapsto w^3$ ("cubing") is the radial segment $[0,\operatorname{cis} 3t]$.
This is because $(r\operatorname{cis}\theta)^n = r^n\operatorname{cis}n\theta$.
So the image of the filled-in region under cubing is the union of the segments $[0,\operatorname{cis} 3t]$ for $\frac{\pi}{12}\leq t\leq \frac{7\pi}{12}$. Said differently, it is the union of the segments $[0,\operatorname{cis} s]$ for $\frac{3\pi}{12}\leq s\leq \frac{21\pi}{12}$, which interval can be written as $\frac{\pi}{4}\leq s\leq \frac{7\pi}{4}$.
Note that the region did not wrap around and meet itself under this map, so the missing point $z$ would not have been covered by another point.
It remains now to remove the image of $z=0.8\operatorname{cis}\frac{\pi}{6}$ under cubing, which is $0.512\operatorname{cis}\frac{\pi}{2}$ (you incorrectly wrote it as $0.512\operatorname{cis}\frac{\pi}{6}$).
