Hipi Zhdripi i Matematikës/1083

Nga formula e Moivrit përftohen formulat:

	$\scriptstyle {\mathsf {\sin {\text{n}}\varphi }}$	$\scriptstyle {\mathsf {={\binom {n}{1}}\sin \varphi ^{n-1}{\binom {n}{3}}\sin ^{3}\varphi \ \cos ^{n-3}\varphi +\dots +}}$
		$\scriptstyle {\mathsf {+(-1)^{{\text{s}}-1 \over 2}{\binom {n}{s}}\sin ^{2}\varphi \ \cos ^{n-s}\varphi }}$ ku $\scriptstyle {\mathsf {s={\Big \{}{\begin{matrix}\scriptstyle {\mathsf {n,\ kur\ n=2k+1}}\\\scriptstyle {\mathsf {n-1,\ kur\ n=2k}}\end{matrix}}}}$	{...30}

dhe

	$\scriptstyle {\mathsf {\cos n\varphi }}$	$\scriptstyle {\mathsf {={\binom {n}{0}}\cos ^{n}\varphi -{\binom {n}{2}}\cos ^{n-s}\varphi \sin ^{2}\varphi +{\binom {n}{4}}\cos ^{n-4}\varphi \ \sin ^{4}\varphi -\dots +}}$
		$\scriptstyle {\mathsf {+(-1)^{s \over 2}{\binom {n}{s}}\cos ^{n-s}\varphi \ \sin ^{s}\varphi }}$ ku $\scriptstyle {\mathsf {s={\Big \{}{\begin{matrix}\scriptstyle {\mathsf {n,\ kur\ n=2k}}\\\scriptstyle {\mathsf {n-1,\ kur\ n=2k+1}}\end{matrix}}}}$	(...31)

S h e m b u l l i 2. - Të njehsohet

\scriptstyle {\mathsf {(1+\cos \theta +i\sin \theta )^{n}}}

ku $\scriptstyle \in$

Z g j i d h j e : Më parë numrin e dhënë e transformojmë në formën trigonometrike:

	$\scriptstyle {\mathsf {\mod (1+\cos \theta +i\sin \theta )}}$	$\scriptstyle {\mathsf {={\sqrt {(1+\cos \theta )^{2}+\sin ^{2}\theta }}={\sqrt {2(1+\cos \theta )}}}}$
		$\scriptstyle {\mathsf {={\sqrt {\textstyle {4\cos ^{2}\theta \over 2}}}=2\cos {\textstyle {\theta \over 2}}}}$ ,

	$\scriptstyle {\mathsf {\arg(1+\cos \theta +i\sin \theta )}}$	$\scriptstyle {\mathsf {=arc\ tg{\textstyle {\sin \theta \over {(1+\cos \theta )}}}=arc\ tg{{sin{\textstyle {\theta \over 2}}} \over {\scriptstyle {\theta }}}}}$
		$\scriptstyle {\mathsf {=arc\,tg\,tg{\textstyle {\theta \over 2}}={\textstyle {\theta \over 2}}}},$

pra, kemi:

\scriptstyle {\mathsf {1+\cos \theta +i\sin \theta =2\cos {\textstyle {\theta  \over 2}}{\big (}\cos {\textstyle {\theta  \over 2}}+i\sin {\textstyle {\theta  \over 2}}{\big )}.}}

Tani, në bazë të formulës (28), përftohet:

	$\scriptstyle {\mathsf {{1+\cos \theta +i\sin \theta }^{n}}}$	$\scriptstyle {\mathsf {={\big [}2\cos {\textstyle {\theta \over 2}}{\big (}\cos {\textstyle {\theta \over 2}}+i\sin {\textstyle {\theta \over 2}}{\big )}{\big ]}^{n}}}$
		$\scriptstyle {\mathsf {=2^{n}\cos {\textstyle {\theta \over 2}}{\big (}\cos {\textstyle {n\theta \over 2}}+i\sin {\textstyle {n\theta \over 2}}{\big )}.}}$