Բանաձևերում
k
{\displaystyle k}
n
{\displaystyle n}
sin
(
π
4
+
x
)
=
cos
(
π
4
−
x
)
.
{\displaystyle \sin \left({\frac {\pi }{4}}+x\right)=\cos \left({\frac {\pi }{4}}-x\right).}
sin
(
π
4
−
x
)
=
cos
(
π
4
+
x
)
.
{\displaystyle \sin \left({\frac {\pi }{4}}-x\right)=\cos \left({\frac {\pi }{4}}+x\right).}
1
±
sin
x
=
2
sin
2
(
π
4
±
x
2
)
.
{\displaystyle 1\pm \sin x=2\sin ^{2}\left({\frac {\pi }{4}}\pm {\frac {x}{2}}\right).}
1
+
cos
x
=
2
cos
2
(
x
2
)
.
{\displaystyle 1+\cos x=2\cos ^{2}\left({\frac {x}{2}}\right).}
1
−
cos
x
=
2
sin
2
(
x
2
)
.
{\displaystyle 1-\cos x=2\sin ^{2}\left({\frac {x}{2}}\right).}
sin
2
x
=
1
1
+
ctg
2
x
.
{\displaystyle \sin ^{2}x={\frac {1}{1+\operatorname {ctg} ^{2}x}}.}
cos
2
x
=
1
1
+
tg
2
x
.
{\displaystyle \cos ^{2}x={\frac {1}{1+\operatorname {tg} ^{2}x}}.}
sin
2
x
−
sin
2
y
=
sin
(
x
−
y
)
⋅
sin
(
x
+
y
)
.
{\displaystyle \sin ^{2}x-\sin ^{2}y=\sin(x-y)\cdot \sin(x+y).}
cos
2
x
−
cos
2
y
=
−
sin
(
x
−
y
)
⋅
sin
(
x
+
y
)
.
{\displaystyle \cos ^{2}x-\cos ^{2}y=-\sin(x-y)\cdot \sin(x+y).}
cos
2
x
−
sin
2
y
=
cos
(
x
−
y
)
⋅
cos
(
x
+
y
)
.
{\displaystyle \cos ^{2}x-\sin ^{2}y=\cos(x-y)\cdot \cos(x+y).}
sin
2
x
+
sin
2
y
=
2
cos
(
x
−
y
)
⋅
sin
(
x
+
y
)
.
{\displaystyle \sin 2x+\sin 2y=2\cos(x-y)\cdot \sin(x+y).}
sin
2
x
−
sin
2
y
=
2
sin
(
x
−
y
)
⋅
cos
(
x
+
y
)
.
{\displaystyle \sin 2x-\sin 2y=2\sin(x-y)\cdot \cos(x+y).}
cos
2
x
+
cos
2
y
=
2
cos
(
x
−
y
)
⋅
cos
(
x
+
y
)
.
{\displaystyle \cos 2x+\cos 2y=2\cos(x-y)\cdot \cos(x+y).}
cos
2
x
−
cos
2
y
=
−
2
sin
(
x
−
y
)
⋅
sin
(
x
+
y
)
.
{\displaystyle \cos 2x-\cos 2y=-2\sin(x-y)\cdot \sin(x+y).}
sin
2
x
+
cos
2
y
=
2
sin
(
π
4
+
x
−
y
)
⋅
sin
(
π
4
+
x
+
y
)
.
{\displaystyle \sin 2x+\cos 2y=2\sin \left({\frac {\pi }{4}}+x-y\right)\cdot \sin \left({\frac {\pi }{4}}+x+y\right).}
sin
2
x
−
cos
2
y
=
−
2
sin
(
π
4
−
x
−
y
)
⋅
sin
(
π
4
−
x
+
y
)
.
{\displaystyle \sin 2x-\cos 2y=-2\sin \left({\frac {\pi }{4}}-x-y\right)\cdot \sin \left({\frac {\pi }{4}}-x+y\right).}
sin
3
x
+
cos
3
x
=
(
sin
x
+
cos
x
)
(
1
−
sin
x
cos
x
)
.
{\displaystyle \sin ^{3}x+\cos ^{3}x=(\sin x+\cos x)(1-\sin x\cos x).}
sin
4
x
+
cos
4
x
=
1
−
2
sin
2
x
cos
2
x
=
1
−
1
2
sin
2
(
2
x
)
=
3
4
+
1
4
cos
(
4
x
)
.
{\displaystyle \sin ^{4}x+\cos ^{4}x=1-2\sin ^{2}x\,\cos ^{2}x=1-{\frac {1}{2}}\sin ^{2}(2x)={\frac {3}{4}}+{\frac {1}{4}}\cos(4x).}
sin
6
x
+
cos
6
x
=
1
−
3
sin
2
x
cos
2
x
=
1
−
3
sin
2
x
+
3
sin
4
x
=
1
−
3
4
sin
2
(
2
x
)
=
5
8
+
3
8
cos
(
4
x
)
.
{\displaystyle \sin ^{6}x+\cos ^{6}x=1-3\sin ^{2}x\,\cos ^{2}x=1-3\sin ^{2}x+3\sin ^{4}x=1-{\frac {3}{4}}\sin ^{2}(2x)={\frac {5}{8}}+{\frac {3}{8}}\cos(4x).}
1
±
tg
x
=
2
sin
(
π
4
±
x
)
cos
x
.
{\displaystyle 1\pm \operatorname {tg} x={\frac {{\sqrt {2}}\sin \left({\frac {\pi }{4}}\pm x\right)}{\cos x}}.}
1
±
ctg
x
=
2
sin
(
π
4
±
x
)
sin
x
.
{\displaystyle 1\pm \operatorname {ctg} x={\frac {{\sqrt {2}}\sin \left({\frac {\pi }{4}}\pm x\right)}{\sin x}}.}
tg
x
=
sin
2
x
cos
2
x
+
1
=
1
−
cos
2
x
sin
2
x
.
{\displaystyle \operatorname {tg} x={\frac {\sin 2x}{\cos 2x+1}}={\frac {1-\cos 2x}{\sin 2x}}.}
ctg
2
x
−
tg
2
x
=
4
cos
2
x
sin
2
2
x
.
{\displaystyle \operatorname {ctg} ^{2}x-\operatorname {tg} ^{2}x={\frac {4\cos 2x}{\sin ^{2}2x}}.}
sin
3
x
=
4
sin
x
⋅
sin
(
π
3
+
x
)
⋅
sin
(
π
3
−
x
)
.
{\displaystyle \sin 3x=4\sin x\cdot \sin \left({\frac {\pi }{3}}+x\right)\cdot \sin \left({\frac {\pi }{3}}-x\right).}
tg
3
x
=
tg
x
⋅
tg
(
π
3
+
x
)
⋅
tg
(
π
3
−
x
)
.
{\displaystyle \operatorname {tg} 3x=\operatorname {tg} x\cdot \operatorname {tg} \left({\frac {\pi }{3}}+x\right)\cdot \operatorname {tg} \left({\frac {\pi }{3}}-x\right).}
sin
5
x
=
16
sin
x
⋅
sin
(
π
5
+
x
)
⋅
sin
(
π
5
−
x
)
⋅
sin
(
2
π
5
+
x
)
⋅
sin
(
2
π
5
−
x
)
.
{\displaystyle \sin 5x=16\sin x\cdot \sin \left({\frac {\pi }{5}}+x\right)\cdot \sin \left({\frac {\pi }{5}}-x\right)\cdot \sin \left({\frac {2\pi }{5}}+x\right)\cdot \sin \left({\frac {2\pi }{5}}-x\right).}
tg
5
x
=
tg
x
⋅
tg
(
π
5
+
x
)
⋅
tg
(
π
5
−
x
)
⋅
tg
(
2
π
5
+
x
)
⋅
tg
(
2
π
5
−
x
)
.
{\displaystyle \operatorname {tg} 5x=\operatorname {tg} x\cdot \operatorname {tg} \left({\frac {\pi }{5}}+x\right)\cdot \operatorname {tg} \left({\frac {\pi }{5}}-x\right)\cdot \operatorname {tg} \left({\frac {2\pi }{5}}+x\right)\cdot \operatorname {tg} \left({\frac {2\pi }{5}}-x\right).}
sin
7
x
=
64
sin
x
⋅
sin
(
π
7
+
x
)
⋅
sin
(
π
7
−
x
)
⋅
sin
(
2
π
7
+
x
)
⋅
sin
(
2
π
7
−
x
)
⋅
sin
(
3
π
7
+
x
)
⋅
sin
(
3
π
7
−
x
)
.
{\displaystyle \sin 7x=64\sin x\cdot \sin \left({\frac {\pi }{7}}+x\right)\cdot \sin \left({\frac {\pi }{7}}-x\right)\cdot \sin \left({\frac {2\pi }{7}}+x\right)\cdot \sin \left({\frac {2\pi }{7}}-x\right)\cdot \sin \left({\frac {3\pi }{7}}+x\right)\cdot \sin \left({\frac {3\pi }{7}}-x\right).}
tg
7
x
=
tg
x
⋅
tg
(
π
7
+
x
)
⋅
tg
(
π
7
−
x
)
⋅
tg
(
2
π
7
+
x
)
⋅
tg
(
2
π
7
−
x
)
⋅
tg
(
3
π
7
+
x
)
⋅
tg
(
3
π
7
−
x
)
.
{\displaystyle \operatorname {tg} 7x=\operatorname {tg} x\cdot \operatorname {tg} \left({\frac {\pi }{7}}+x\right)\cdot \operatorname {tg} \left({\frac {\pi }{7}}-x\right)\cdot \operatorname {tg} \left({\frac {2\pi }{7}}+x\right)\cdot \operatorname {tg} \left({\frac {2\pi }{7}}-x\right)\cdot \operatorname {tg} \left({\frac {3\pi }{7}}+x\right)\cdot \operatorname {tg} \left({\frac {3\pi }{7}}-x\right).}
sin
n
x
=
2
n
−
1
∏
k
=
0
n
−
1
sin
(
x
+
π
k
n
)
.
{\displaystyle \sin nx=2^{n-1}\prod \limits _{k=0}^{n-1}\sin \left(x+{\frac {\pi k}{n}}\right).}
tg
[
(
2
n
+
1
)
x
]
=
(
−
1
)
n
∏
k
=
0
2
n
tg
(
x
+
π
k
2
n
+
1
)
.
{\displaystyle \operatorname {tg} {\big [}(2n+1)x{\big ]}=(-1)^{n}\prod \limits _{k=0}^{2n}\operatorname {tg} \left(x+{\frac {\pi k}{2n+1}}\right).}
∑
k
=
1
n
sin
(
k
x
)
=
sin
(
n
+
1
2
x
)
sin
(
n
x
2
)
sin
(
x
2
)
.
{\displaystyle \sum \limits _{k=1}^{n}\sin(kx)=\sin \left({\frac {n+1}{2}}x\right){\frac {\sin \left({\frac {nx}{2}}\right)}{\sin \left({\frac {x}{2}}\right)}}.}
∑
k
=
1
n
cos
(
k
x
)
=
cos
(
n
+
1
2
x
)
sin
(
n
x
2
)
sin
(
x
2
)
.
{\displaystyle \sum \limits _{k=1}^{n}\cos(kx)=\cos \left({\frac {n+1}{2}}x\right){\frac {\sin \left({\frac {nx}{2}}\right)}{\sin \left({\frac {x}{2}}\right)}}.}
∑
k
=
1
n
sin
[
(
2
k
−
1
)
x
]
=
sin
2
n
x
sin
x
.
{\displaystyle \sum \limits _{k=1}^{n}\sin {\big [}(2k-1)x{\big ]}={\frac {\sin ^{2}nx}{\sin x}}.}
∑
k
=
1
n
cos
[
(
2
k
−
1
)
x
]
=
sin
2
n
x
2
sin
x
.
{\displaystyle \sum \limits _{k=1}^{n}\cos {\big [}(2k-1)x{\big ]}={\frac {\sin 2nx}{2\sin x}}.}
∑
k
=
1
n
cos
2
π
k
2
n
+
1
=
−
1
2
.
{\displaystyle \sum \limits _{k=1}^{n}\cos {\frac {2\pi k}{2n+1}}=-{\frac {1}{2}}.}
∏
k
=
1
n
−
1
sin
k
π
n
=
n
2
n
−
1
.
{\displaystyle \prod \limits _{k=1}^{n-1}\sin {\frac {k\pi }{n}}={\frac {n}{2^{n-1}}}.}
∏
k
=
1
n
sin
k
π
2
(
n
+
1
)
=
n
+
1
2
n
.
{\displaystyle \prod \limits _{k=1}^{n}\sin {\frac {k\pi }{2(n+1)}}={\frac {\sqrt {n+1}}{2^{n}}}.}
∏
k
=
1
n
sin
k
π
2
n
+
1
=
2
n
+
1
2
n
.
{\displaystyle \prod \limits _{k=1}^{n}\sin {\frac {k\pi }{2n+1}}={\frac {\sqrt {2n+1}}{2^{n}}}.}
∏
k
=
1
2
n
−
1
cos
k
π
n
=
(
−
1
)
n
−
1
2
2
n
−
1
.
{\displaystyle \prod \limits _{k=1}^{2n-1}\cos {\frac {k\pi }{n}}={\frac {(-1)^{n}-1}{2^{2n-1}}}.}
∏
k
=
0
n
cos
(
2
k
x
)
=
sin
(
2
n
+
1
x
)
2
n
+
1
sin
x
.
{\displaystyle \prod \limits _{k=0}^{n}\cos \left(2^{k}x\right)={\frac {\sin \left(2^{n+1}x\right)}{2^{n+1}\sin x}}.}
∏
k
=
0
n
cos
x
2
k
=
sin
2
x
2
n
+
1
sin
(
x
2
n
)
.
{\displaystyle \prod \limits _{k=0}^{n}\cos {\frac {x}{2^{k}}}={\frac {\sin 2x}{2^{n+1}\sin \left({\frac {x}{2^{n}}}\right)}}.}
∏
k
=
1
n
cos
x
2
k
=
sin
x
2
n
sin
(
x
2
n
)
.
{\displaystyle \prod \limits _{k=1}^{n}\cos {\frac {x}{2^{k}}}={\frac {\sin x}{2^{n}\sin \left({\frac {x}{2^{n}}}\right)}}.}
∏
k
=
0
∞
cos
x
2
k
=
sin
2
x
2
x
.
{\displaystyle \prod \limits _{k=0}^{\infty }\cos {\frac {x}{2^{k}}}={\frac {\sin 2x}{2x}}.}
∏
k
=
1
∞
cos
x
2
k
=
sin
x
x
.
{\displaystyle \prod \limits _{k=1}^{\infty }\cos {\frac {x}{2^{k}}}={\frac {\sin x}{x}}.}
cos
20
∘
⋅
cos
40
∘
⋅
cos
80
∘
=
1
8
.
{\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}}.}
cos
π
7
⋅
cos
4
π
7
⋅
cos
5
π
7
=
1
8
.
{\displaystyle \cos {\frac {\pi }{7}}\cdot \cos {\frac {4\pi }{7}}\cdot \cos {\frac {5\pi }{7}}={\frac {1}{8}}.}
cos
π
7
⋅
cos
2
π
7
⋅
cos
4
π
7
=
−
1
8
.
{\displaystyle \cos {\frac {\pi }{7}}\cdot \cos {\frac {2\pi }{7}}\cdot \cos {\frac {4\pi }{7}}=-{\frac {1}{8}}.}
cos
π
9
⋅
cos
2
π
9
⋅
cos
4
π
9
=
1
8
.
{\displaystyle \cos {\frac {\pi }{9}}\cdot \cos {\frac {2\pi }{9}}\cdot \cos {\frac {4\pi }{9}}={\frac {1}{8}}.}
cos
π
9
⋅
cos
5
π
9
⋅
cos
7
π
9
=
1
8
.
{\displaystyle \cos {\frac {\pi }{9}}\cdot \cos {\frac {5\pi }{9}}\cdot \cos {\frac {7\pi }{9}}={\frac {1}{8}}.}
cos
24
∘
+
cos
48
∘
+
cos
96
∘
+
cos
168
∘
=
1
2
.
{\displaystyle \cos 24^{\circ }+\cos 48^{\circ }+\cos 96^{\circ }+\cos 168^{\circ }={\frac {1}{2}}.}
cos
(
2
π
21
)
+
cos
(
2
⋅
2
π
21
)
+
cos
(
4
⋅
2
π
21
)
+
cos
(
5
⋅
2
π
21
)
+
cos
(
8
⋅
2
π
21
)
+
cos
(
10
⋅
2
π
21
)
=
1
2
.
{\displaystyle {\begin{aligned}&\cos \left({\frac {2\pi }{21}}\right)+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)\\[10pt]&{}\qquad {}+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.\end{aligned}}}
Следующая формула приводится в двух вариантах для угла
α
{\displaystyle \alpha }
tg
α
=
360
∘
⋅
α
π
∑
k
=
1
∞
1
(
180
∘
k
−
90
∘
+
α
)
(
180
∘
k
−
90
∘
−
α
)
=
2
α
∑
k
=
1
∞
1
(
k
−
1
/
2
)
2
π
2
−
α
2
.
{\displaystyle \operatorname {tg} \alpha ={\frac {360^{\circ }\cdot \alpha }{\pi }}\sum \limits _{k=1}^{\infty }{\frac {1}{(180^{\circ }k-90^{\circ }+\alpha )(180^{\circ }k-90^{\circ }-\alpha )}}=2\alpha \sum \limits _{k=1}^{\infty }{\frac {1}{(k-1/2)^{2}\pi ^{2}-\alpha ^{2}}}.}
arctg
1
2
+
arctg
1
3
=
π
4
.
{\displaystyle \operatorname {arctg} {\frac {1}{2}}+\operatorname {arctg} {\frac {1}{3}}={\frac {\pi }{4}}.}
![{\displaystyle \operatorname {tg} {\big [}(2n+1)x{\big ]}=(-1)^{n}\prod \limits _{k=0}^{2n}\operatorname {tg} \left(x+{\frac {\pi k}{2n+1}}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9350c4eb2ccd9ab4805de3dbc37d5ba710932a)
![{\displaystyle \sum \limits _{k=1}^{n}\sin {\big [}(2k-1)x{\big ]}={\frac {\sin ^{2}nx}{\sin x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/406902955ce267485d9fd04835e1eefc296a9b13)
![{\displaystyle \sum \limits _{k=1}^{n}\cos {\big [}(2k-1)x{\big ]}={\frac {\sin 2nx}{2\sin x}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09ebe4c43a1205e9d2b21f64365f220b27cdd981)
![{\displaystyle {\begin{aligned}&\cos \left({\frac {2\pi }{21}}\right)+\cos \left(2\cdot {\frac {2\pi }{21}}\right)+\cos \left(4\cdot {\frac {2\pi }{21}}\right)\\[10pt]&{}\qquad {}+\cos \left(5\cdot {\frac {2\pi }{21}}\right)+\cos \left(8\cdot {\frac {2\pi }{21}}\right)+\cos \left(10\cdot {\frac {2\pi }{21}}\right)={\frac {1}{2}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/15e07fe52f1162c9c933e28a6bec4fb9f59528e8)
