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 )}.}}}
![{\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)
