Question

Find all the values of the given root :
$\sqrt{1+i}$

Answer 1

$1\pm i=\sqrt{2}(cos\frac{\pi }{4}\pm isin\frac{\pi }{4})$
$=\sqrt{2}[cos(2n\pi +\frac{\pi }{4})\pm isin(2n\pi +\frac{\pi }{4})]$
$=\sqrt{2}\left[cos\right(8n+1)\frac{\pi }{4}\pm isin(8n+1\left)\frac{\pi }{4}\right]$
$\therefore \sqrt{(1\pm i)}=2^{1/4}\left[cos\right(8n+1)\frac{\pi }{8}\pm isin(8n+1\left)\frac{\pi }{8}\right]$
where n = 0,1
$=2^{1/4}[cos\frac{\pi }{8}\pm isin\frac{\pi }{8}]forn=0$
$=2^{1/4}[cos(\pi +\frac{\pi }{8})\pm isin(\pi +\frac{\pi }{8})]forn=0$
$=-2^{1/4}(cos\frac{\pi }{8}\pm isin\frac{\pi }{8})where$
$cos=\sqrt{\left(\frac{\sqrt{2}+1}{2\sqrt{2}}\right)},sin=\sqrt{\left(\frac{\sqrt{2}-1}{2\sqrt{2}}\right)}$