So I have these questions...
and these are the answers...
Now I can see that they have used the angle $\frac{\pi}{2}$ where $\theta$ goes in $e^{i\theta}$ but otherwise I can't see the logic in this, I haven't been given any formulas to say what $z$ or $z'$ is equal to? Are they using $z=re^{i\theta}$?
Sorry if this is basic stuff but this lecturer struggles to even get that across to us?
To get you started ...
You probably know that any complex number can be written in "polar" form: $z = re^{i\theta}$. Then, if $z = pe^{i\alpha}$ and $w = qe^{i\beta}$, we have $$ zw = (pq)e^{i(\alpha+\beta)} $$ Further explanation given here. So, in particular, suppose $z = re^{i\alpha}$, and consider the complex number $$ z' = z \, e^{i\theta} = r e^{i(\alpha+\theta)} $$ We see that $z'$ has the same magnitude as the original $z$, but $\arg(z') = \alpha +\theta = \arg(z) + \theta$. In other words, $z'$ is obtained by rotating $z$ through an angle $\theta$ around the origin. This means that the mapping $z \mapsto z \, e^{i\theta}$ performs a rotation by an angle $\theta$ around the origin.