Question

The argument of $\frac{1-i}{1+i}$ is
(a) $-\frac{\mathrm{\pi }}{2}$
(b) $\frac{\mathrm{\pi }}{2}$
(c) $\frac{3\mathrm{\pi }}{2}$
(d) $\frac{5\mathrm{\pi }}{2}$

Answer 1

(a) $-\frac{\pi }{2}$

$$\begin{gathered} \mathrm{Let}z=\frac{1-i}{1+i} \\ \Rightarrow z=\frac{1-i}{1+i}\times \frac{1-i}{1-i} \\ \Rightarrow z=\frac{1+i^2-2i}{1-i^2} \\ \Rightarrow z=\frac{1-1-2i}{1+1} \\ \Rightarrow z=\frac{-2i}{2} \\ \Rightarrow z=-i \\ \mathrm{Since},z\mathrm{lies}\mathrm{on}\mathrm{negative}\mathrm{direction}\mathrm{of}\mathrm{imaginary}\mathrm{axis}. \\ \mathrm{Therefore},\arg \left(z\right)=\frac{-\mathrm{\pi }}{2} \end{gathered}$$