Question

Perform the indicated operations.
$(1+cos\varphi +isin\varphi {)}^n.$

Answer 1

${(1+\cos \varphi +i\sin \varphi )}^n$

$(1+\cos \varphi )+i\sin \varphi$

$\tan \theta =\frac{\sin \varphi }{1+\cos \varphi }$

$r=\sqrt{{(1+\cos \varphi )}^2+{\sin }^2\varphi }$

$\Rightarrow \sqrt{1+{\cos }^2\varphi +2\cos \varphi +{\sin }^2\varphi }$

$\Rightarrow \sqrt{2}\sqrt{1+\cos \varphi }$

$1+\cos \varphi +i\sin \varphi =\sqrt{2}\sqrt{1+\cos \varphi }{e}^{i\theta }$

${(1+\cos \varphi +i\sin \varphi )}^n=\sqrt{2}\sqrt{1+\cos \varphi }{e}^{in\theta }$