I have been asked to solve the following and represent the answer graphically:
A) $| \arg z - (\pi/4) | < (\pi/2)$
I understand that this means the difference between the argument of $z$ and $(\pi/4)$ has to lie in the first quadrant, but I'm not quite certain how to represent this on a graph.
B) $| \arg ( 1+ i )z| \leq (\pi/2)$.
This one and the next are completely lost on me.
C) $| \arg z - \arg (1+i)| < (\pi/2)$.
I assume here $\arg (1+i)$ would be $\tan^{-}1 (1/1) = (\pi/4)$ and that would make it exactly like the first problem?
D) $|z-i|/|z+i| \leq 1$.
I know how to solve this one, but I'm not quite sure what area (above or below the perpendicular bisector of the line joining -i and i) I'm required to shade/mark. An explanation as to how I should figure this out would be appreciated!
Thanks a lot in advance for all your help!
Hint: One way to approach these sorts of questions is to try and think geometrically about what the equation is saying. For example, for D, rearranging gives $|z-i|\leq|z+i|$, which in words says '$z$ is closer to $i$ than to $-i$'.