sin n φ = ( n 1 ) sin φ n − 1 ( n 3 ) sin 3 φ cos n − 3 φ + ⋯ + {\displaystyle \scriptstyle {\mathsf {\sin {\text{n}}\varphi ={\binom {\text{n}}{1}}\sin \varphi ^{{\text{n}}-1}{\binom {\text{n}}{3}}\sin ^{3}\varphi \ \cos ^{{\text{n}}-3}\varphi +\dots +}}}
+ ( − 1 ) s − 1 2 ( n s ) sin 2 φ cos n-s φ k u s = { n, kur n=2k+1 n-1, kur n=2k {\displaystyle \scriptstyle {\mathsf {+(-1)^{{\text{s}}-1 \over 2}{\binom {\text{n}}{\text{s}}}\sin ^{2}\varphi \ \cos ^{\text{n-s}}\varphi \ ku\ s={\Big \{}{\begin{matrix}\scriptstyle {\mathsf {\text{n, kur n=2k+1}}}\\\scriptstyle {\mathsf {\text{n-1, kur n=2k }}}\end{matrix}}}}}
cos n φ = ( n 0 ) cos n φ − ( n 2 ) cos n-s φ sin 2 φ + ( n 4 ) cos n-4 φ sin 4 φ − ⋯ + {\displaystyle \scriptstyle {\mathsf {\cos {\text{n}}\varphi ={\binom {\text{n}}{0}}\cos ^{\text{n}}\varphi -{\binom {\text{n}}{2}}\cos ^{\text{n-s}}\varphi \sin ^{2}\varphi +{\binom {\text{n}}{4}}\cos ^{\text{n-4}}\varphi \ \sin ^{4}\varphi -\dots +}}}
+ ( − 1 ) s 2 ( n s ) cos n-s φ sin s φ k u s = { n, kur n=2k n-1, kur n=2k+1 {\displaystyle \scriptstyle {\mathsf {+(-1)^{{\text{s}} \over 2}{\binom {\text{n}}{\text{s}}}\cos ^{\text{n-s}}\varphi \ \sin ^{\text{s}}\varphi \ ku\ s={\Big \{}{\begin{matrix}\scriptstyle {\mathsf {\text{n, kur n=2k}}}\\\scriptstyle {\mathsf {\text{n-1, kur n=2k+1 }}}\end{matrix}}}}}
( 1 + cos θ + i sin θ ) n ku n ∈ N . {\displaystyle \scriptstyle {\mathsf {(1+\cos \theta +i\sin \theta )^{n}\ {\text{ku}}\ n\in \mathbb {N} .}}}
mod ( 1 + cos θ + i sin θ ) = ( 1 + cos θ ) 2 + sin 2 θ = 2 ( 1 + cos θ ) {\displaystyle \scriptstyle {\mathsf {\mod (1+\cos \theta +i\sin \theta )={\sqrt {(1+\cos \theta )^{2}+\sin ^{2}\theta }}={\sqrt {2(1+\cos \theta )}}}}}
= 4 cos 2 θ 2 = 2 cos θ 2 {\displaystyle \scriptstyle {\mathsf {={\sqrt {4\cos ^{2}\theta \over 2}}=2\cos {\theta \over 2}}}} ,
arg ( 1 + cos θ + i sin θ ) = a r c t g sin θ ( 1 + cos θ ) = a r c t g s i n θ 2 θ {\displaystyle \scriptstyle {\mathsf {\arg(1+\cos \theta +i\sin \theta )=arc\ tg{\sin \theta \over {(1+\cos \theta )}}=arc\ tg{{sin{\theta \over 2}} \over \theta }}}}
= a r c t g t g θ 2 = θ 2 , {\displaystyle \scriptstyle {\mathsf {=arc\,tg\,tg{\theta \over 2}={\theta \over 2}}},}
1 + cos θ + i sin θ = 2 cos θ 2 ( cos θ 2 + i sin θ 2 ) . {\displaystyle \scriptstyle {\mathsf {1+\cos \theta +i\sin \theta =2\cos {\theta \over 2}{\big (}\cos {\theta \over 2}+i\sin {\theta \over 2}{\big )}.}}}
1 + cos θ + i sin θ n = [ 2 cos θ 2 ( cos θ 2 + i sin θ 2 ) ] n {\displaystyle \scriptstyle {\mathsf {{1+\cos \theta +i\sin \theta }^{n}={\big [}2\cos {\theta \over 2}{\big (}\cos {\theta \over 2}+i\sin {\theta \over 2}{\big )}{\big ]}^{n}}}}
= 2 n cos θ 2 ( cos n θ 2 + i sin n θ 2 ) . {\displaystyle \scriptstyle {\mathsf {=2^{n}\cos {\theta \over 2}{\big (}\cos {n\theta \over 2}+i\sin {n\theta \over 2}{\big )}.}}}
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![{\displaystyle \scriptstyle {\mathsf {{1+\cos \theta +i\sin \theta }^{n}={\big [}2\cos {\theta \over 2}{\big (}\cos {\theta \over 2}+i\sin {\theta \over 2}{\big )}{\big ]}^{n}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8469ef54e90d3fdf746966826dd7ab0a9e0c844a)
