Think geometrically. Do you know about Euler's formula? It states that
$$e^{i\theta} = \cos\theta + i \sin\theta$$
By comparing, you can see that your number is $w = e^{2\pi i/5}$, which geometrically corresponds to the point in the argand diagram a distance 1 from the origin, with argument $2\pi/5$, which is 1/5 of the way round a circle.
Squaring to get $w^2$ leaves the modulus unchanged and doubles the argument, so the point $w^2$ is 2/5 of the way round the circle, and $w^3$ and $w^4$ are 3/5 and 4/5 of the way round the circle.
Try plotting the points $1, w, w^2, w^3, w^4$ on the argand diagram and see what you notice about them. Now can you start adding them up?
For your first part, let $z=x+iy$ so then we get that
$$
| z- (\sqrt{3} + i)| = | (x-\sqrt{3}) + i(y-1)| = (x-\sqrt{3})^2 + (y-1)^2 = 2
$$
which is the locus of a circle with centre $(\sqrt{3},1)$ and radius $ \sqrt{2}$. Substituting $ x= 0, y=0$ gives us that it intersects the origin.
For your second part, consider that arg z = $arctan(\frac{y}{x})$.
Best Answer
Could you not just plot $$z=\theta(\cos\theta+i\cdot\sin\theta)$$ The typical form is $z=r(\cos\theta+i\cdot\sin\theta)$ but given $|z|=r$ and $arg(z)=\theta$, for your case $r=\theta$