Question

If |z| = 2 and arg (z) = $\frac{\mathrm{\pi }}{4}$, then z = ____________.

Answer 1

for |z| = 2 = r arg z = $\frac{\mathrm{\pi }}{4}$

z = r (cos(arg z) + i sin (arg z))

$$\begin{gathered} \mathrm{i}.\mathrm{e}z=2\left(\cos \frac{\mathrm{\pi }}{4}+i\sin \frac{\mathrm{\pi }}{4}\right) \\ =2\left(\frac{1}{\sqrt{2}}+i\frac{1}{\sqrt{2}}\right) \\ z=\sqrt{2}+i\sqrt{2} \\ \mathrm{hence},z=\sqrt{2}\left(1+i\right) \end{gathered}$$