For an argument form of a half line where $$ arg(z - a - bi) = π/2 $$ I can see that the cartesian equation will be $$ x = a $$ As a line of angle π/2 radians will be vertical.
How do I best show the working for this? Do I simply say, tan(π/2) is undefined and will result in a vertical line, or can I say that $$ tan(π/2) = 1/0 $$ as $$ tan(x) = sin(x)/cos(x) $$ Then from this I get the lines of working: $$ (y-b)/(x-a) = 1/0 \\ x-a = 0 \\ x = a $$ But then this could also be rearranged to give y = ∞
In order that $\arg(w)=\frac{\pi}{2}$ we require that $$\Re(w)=0$$ and $$\Im(w)>0$$
So in this case, we have $x-a=0$ and $y-b>0$