Question

Find the arg(z),if z=
$\frac{1}{1 + i}$
​

Answer 1

Answer:

your solution is as follows

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Step-by-step explanation:

$to \: find : \\ arg ument\: of \: complex \: number \\ arg \: (z) \\ \\ given \: complex \: number \: is \\ z = \frac{1}{1 + i} \\ \\ = \frac{1}{1 + i} \times \frac{1 - i}{1 - i} \\ \\ = \frac{1 - i}{(1 + i)(1 - i)} \\ \\ = \frac{1 - i}{1 - i {}^{2} } \\ \\ = \frac{1 - i}{1 - ( - 1)} \\ \\ = \frac{1 - i}{2} \\ \\ equating \: real \: and \: imaginary \: parts \\ we \: get \\ \\ a = \frac{1}{2} \: \: , \: b = - \frac{1}{2}$

$we \: know \: that \\ arg \: (z) = tan \: x = \frac{b}{a} \\ \\ = \frac{ \frac{ \frac{ - 1}{2} }{1} }{2} \\ \\ = \frac{ - 1}{1} \\ \\ = - 1 \\ \\ we \: know \: that \\ tan \: \: \frac{\pi}{4} = 1 \\ \\ thus \: then \: \\ x = \frac{\pi}{4}$

$but \: since \: here \\ (a,b) = ( \frac{1}{2} , - \frac{1}{2} \: ) \: \: lies \: in \: 4th \: quadrant \\ \\ we \: know \: that \\ angle \: in \: fourth \: quadrant \\ is \: calculated \: as \\ 2\pi - x \\ = 2\pi - \frac{\pi}{4} \\ \\ = \frac{8\pi - \pi}{4} \\ \\arg \: (z) = \frac{7\pi {}^{c} }{4}$