Describe the set of points z in the complex plane that satisfy $\displaystyle arg(z)=\frac{\pi}{4}$.
Is this question just asking for $\displaystyle x=\frac{\sqrt{2}}{2}$ and $\displaystyle y=\frac{\sqrt{2}}{2}$?
no, the set is a whole line, not just one point. All complex numbers on the line $\displaystyle r(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}\cdot i)=r e^{i \frac{\pi}{4}}$ for r $\displaystyle \in \mathbb{R}, r>0$.
See also
Complex number - Wikipedia, the free encyclopedia