Тригонометричні тотожності — математичні вирази з тригонометричними функціями , що виконуються для всіх значень аргумента зі спільної області визначення.
В цій статті кути позначені грецькими буквами
α
,
β
,
γ
{\displaystyle \alpha ,\beta ,\gamma }
градусах або радіанах :
1 повне коло = 360 градусів = 2
π
{\displaystyle \pi }
В наступній таблиці наведено спвівідношення між значеннями в градусах і радіанах для деяких кутів
Градуси
30°
60°
120°
150°
210°
240°
300°
330°
Радіани
π
6
{\displaystyle {\frac {\pi }{6}}\!}
π
3
{\displaystyle {\frac {\pi }{3}}\!}
2
π
3
{\displaystyle {\frac {2\pi }{3}}\!}
5
π
6
{\displaystyle {\frac {5\pi }{6}}\!}
7
π
6
{\displaystyle {\frac {7\pi }{6}}\!}
4
π
3
{\displaystyle {\frac {4\pi }{3}}\!}
5
π
3
{\displaystyle {\frac {5\pi }{3}}\!}
11
π
6
{\displaystyle {\frac {11\pi }{6}}\!}
Градуси
45°
90°
135°
180°
225°
270°
315°
360°
Радіани
π
4
{\displaystyle {\frac {\pi }{4}}\!}
π
2
{\displaystyle {\frac {\pi }{2}}\!}
3
π
4
{\displaystyle {\frac {3\pi }{4}}\!}
π
{\displaystyle \pi \!}
5
π
4
{\displaystyle {\frac {5\pi }{4}}\!}
3
π
2
{\displaystyle {\frac {3\pi }{2}}\!}
7
π
4
{\displaystyle {\frac {7\pi }{4}}\!}
2
π
{\displaystyle 2\pi \!}
Якщо не сказано інакше, то всі кути задано у радіанах, а кути, що закінчуються символом (°) — в градусах.
ред.
У статті будуть наведені співвідношення та тотожності для шести основних тригонометричних функцій:
синус
sin
α
,
{\displaystyle \sin \alpha ,}
косинус
cos
α
,
{\displaystyle \cos \alpha ,}
тангенс
tg
α
=
sin
α
cos
α
,
α
≠
π
2
+
π
n
,
n
∈
Z
,
{\displaystyle \operatorname {tg} \alpha ={\frac {\sin \alpha }{\cos \alpha }},\quad \alpha \neq {\frac {\pi }{2}}+\pi n,\,n\in \mathbb {Z} ,}
котангенс
ctg
α
=
cos
α
sin
α
,
α
≠
π
n
,
n
∈
Z
,
{\displaystyle \operatorname {ctg} \alpha ={\frac {\cos \alpha }{\sin \alpha }},\quad \alpha \neq \pi n,\,n\in \mathbb {Z} ,}
секанс
sec
α
=
1
cos
α
,
α
≠
π
2
+
π
n
,
n
∈
Z
,
{\displaystyle \operatorname {sec} \alpha ={\frac {1}{\cos \alpha }},\quad \alpha \neq {\frac {\pi }{2}}+\pi n,\,n\in \mathbb {Z} ,}
косеканс
csc
α
=
1
sin
α
,
α
≠
π
n
,
n
∈
Z
,
{\displaystyle \operatorname {csc} \alpha ={\frac {1}{\sin \alpha }},\quad \alpha \neq \pi n,\,n\in \mathbb {Z} ,}
В англомовній літературі тангенс та котангенс зазвичай позначають
tan
α
{\displaystyle \tan \alpha }
cot
α
,
{\displaystyle \cot \alpha ,}
ред.
Обернені тригонометричні функції це такі функції, композиція яких зі звичайними тригонометричними функціями дає тотожне відображення. Наприклад, функція обернена до синуса, відома як обернений синус (sin−1 ) або арксинус (arcsin or asin), задовольняє співвідношення
sin
(
arcsin
x
)
=
x
,
|
x
|
≤
1
{\displaystyle \sin(\arcsin x)=x,\quad \quad |x|\leq 1}
та
arcsin
(
sin
x
)
=
x
,
|
x
|
≤
π
2
.
{\displaystyle \arcsin(\sin x)=x,\quad \quad |x|\leq {\frac {\pi }{2}}.}
Тригонометричні функції та обернені до них наведені в наступній таблиці:
Функція
sin
cos
tg
ctg
sec
csc
Обернена
arcsin
arccos
arctg
arcctg
arcsec
arccsc
ред.
Крім основних шести, також використовують інші тригонометричні функції кута. Їх використовували раніше при розв'язуванні різних навігаційних задач, однак з розвитком обчислювальної техніки вони втратили свою актуальність.
Назва
Скорочене позн.
Значення
синус-верзус
versin
(
θ
)
{\displaystyle \operatorname {versin} (\theta )}
vers
(
θ
)
{\displaystyle \operatorname {vers} (\theta )}
ver
(
θ
)
{\displaystyle \operatorname {ver} (\theta )}
1
−
cos
(
θ
)
{\displaystyle 1-\cos(\theta )}
косинус-верзус
vercosin
(
θ
)
{\displaystyle \operatorname {vercosin} (\theta )}
1
+
cos
(
θ
)
{\displaystyle 1+\cos(\theta )}
коверсинус
coversin
(
θ
)
{\displaystyle \operatorname {coversin} (\theta )}
cvs
(
θ
)
{\displaystyle \operatorname {cvs} (\theta )}
1
−
sin
(
θ
)
{\displaystyle 1-\sin(\theta )}
коверкосинус
covercosin
(
θ
)
{\displaystyle \operatorname {covercosin} (\theta )}
1
+
sin
(
θ
)
{\displaystyle 1+\sin(\theta )}
гаверсинус
haversin
(
θ
)
{\displaystyle \operatorname {haversin} (\theta )}
1
−
cos
(
θ
)
2
{\displaystyle {\frac {1-\cos(\theta )}{2}}}
гаверкосинус
havercosin
(
θ
)
{\displaystyle \operatorname {havercosin} (\theta )}
1
+
cos
(
θ
)
2
{\displaystyle {\frac {1+\cos(\theta )}{2}}}
когаверсинус
hacoversin
(
θ
)
{\displaystyle \operatorname {hacoversin} (\theta )}
1
−
sin
(
θ
)
2
{\displaystyle {\frac {1-\sin(\theta )}{2}}}
когаверкосинус
hacovercosin
(
θ
)
{\displaystyle \operatorname {hacovercosin} (\theta )}
1
+
sin
(
θ
)
2
{\displaystyle {\frac {1+\sin(\theta )}{2}}}
ексеканс
exsec
(
θ
)
{\displaystyle \operatorname {exsec} (\theta )}
sec
(
θ
)
−
1
{\displaystyle \sec(\theta )-1}
екскосеканс
excsc
(
θ
)
{\displaystyle \operatorname {excsc} (\theta )}
csc
(
θ
)
−
1
{\displaystyle \csc(\theta )-1}
хорда
crd
(
θ
)
{\displaystyle \operatorname {crd} (\theta )}
2
sin
θ
2
{\displaystyle 2\sin {\frac {\theta }{2}}}
ред.
ред.
ред.
Візуалізація формули (6)
Формули для суми аргументів
sin
(
α
±
β
)
=
sin
α
cos
β
±
cos
α
sin
β
{\displaystyle \sin \left(\alpha \pm \beta \right)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta }
(5)
cos
(
α
±
β
)
=
cos
α
cos
β
∓
sin
α
sin
β
{\displaystyle \cos \left(\alpha \pm \beta \right)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta }
(6)
t
g
(
α
±
β
)
=
t
g
α
±
t
g
β
1
∓
t
g
α
t
g
β
{\displaystyle \mathop {\mathrm {tg} } \left(\alpha \pm \beta \right)={\frac {\mathop {\mathrm {tg} } \alpha \pm \mathop {\mathrm {tg} } \beta }{1\mp \mathop {\mathrm {tg} } \alpha \mathop {\mathrm {tg} } \beta }}}
(7)
ctg
(
α
±
β
)
=
c
t
g
α
c
t
g
β
∓
1
c
t
g
α
±
c
t
g
β
{\displaystyle \operatorname {ctg} \left(\alpha \pm \beta \right)={\frac {\mathop {\mathrm {ctg} } \alpha \mathop {\mathrm {ctg} } \beta \mp 1}{\mathop {\mathrm {ctg} } \alpha \pm \mathop {\mathrm {ctg} } \beta }}}
Формула (7) отримана діленням (5) на (6) .
ред.
sin
(
∑
i
=
1
∞
α
i
)
=
∑
k
=
0
∞
(
−
1
)
k
∑
A
⊆
{
1
,
2
,
3
,
…
}
|
A
|
=
2
k
+
1
(
∏
i
∈
A
sin
α
i
∏
i
∉
A
cos
α
i
)
,
{\displaystyle \sin \left(\sum _{i=1}^{\infty }\alpha _{i}\right)=\sum _{k=0}^{\infty }(-1)^{k}\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=2k+1\end{smallmatrix}}\left(\prod _{i\in A}\sin \alpha _{i}\prod _{i\not \in A}\cos \alpha _{i}\right),}
cos
(
∑
i
=
1
∞
α
i
)
=
∑
k
=
0
∞
(
−
1
)
k
∑
A
⊆
{
1
,
2
,
3
,
…
}
|
A
|
=
2
k
(
∏
i
∈
A
sin
α
i
∏
i
∉
A
cos
α
i
)
.
{\displaystyle \cos \left(\sum _{i=1}^{\infty }\alpha _{i}\right)=\sum _{k=0}^{\infty }~(-1)^{k}~~\sum _{\begin{smallmatrix}A\subseteq \{\,1,2,3,\dots \,\}\\\left|A\right|=2k\end{smallmatrix}}\left(\prod _{i\in A}\sin \alpha _{i}\prod _{i\not \in A}\cos \alpha _{i}\right).}
У правих частинах рівності суму взято по всіх підмножинах натуральних чисел з 2k+1 або 2k елементів відповідно.
ред.
Нехай
e
k
=
e
k
(
x
1
,
…
,
x
n
)
,
k
=
0
,
1
,
2
,
…
,
n
=
1
,
2
,
3
…
,
{\displaystyle e_{k}=e_{k}(x_{1},\ldots ,x_{n}),\,k=0,1,2,\ldots ,n=1,2,3\ldots ,}
елементарні симетричні многочлени степеня k від n змінних
x
i
=
tg
α
i
i
=
1
,
2
,
…
,
n
.
{\displaystyle x_{i}=\operatorname {tg} \alpha _{i}\quad i=1,2,\ldots ,n.}
Наприклад:
e
0
=
1
,
{\displaystyle e_{0}=1,}
e
1
=
∑
i
=
1
n
x
i
=
∑
i
tg
α
i
,
{\displaystyle e_{1}=\sum _{i=1}^{n}x_{i}=\sum _{i}\operatorname {tg} \alpha _{i},}
e
2
=
∑
1
≤
i
<
j
≤
n
x
i
x
j
=
∑
i
<
j
tg
α
i
tg
α
j
,
{\displaystyle e_{2}=\sum _{1\leq i<j\leq n}x_{i}x_{j}=\sum _{i<j}\operatorname {tg} \alpha _{i}\operatorname {tg} \alpha _{j},}
e
3
=
∑
1
≤
i
<
j
<
k
≤
n
x
i
x
j
x
k
=
∑
i
<
j
<
k
tg
α
i
tg
α
j
tg
α
k
.
{\displaystyle e_{3}=\sum _{1\leq i<j<k\leq n}x_{i}x_{j}x_{k}=\sum _{i<j<k}\operatorname {tg} \alpha _{i}\operatorname {tg} \alpha _{j}\operatorname {tg} \alpha _{k}.}
Тоді
tg
(
∑
i
=
1
2
k
α
i
)
=
e
1
−
e
3
+
e
5
−
⋯
+
(
−
1
)
k
+
1
e
2
k
−
1
e
0
−
e
2
+
e
4
−
⋯
+
(
−
1
)
k
e
2
k
=
∑
i
=
1
k
(
−
1
)
i
+
1
e
2
i
−
1
∑
i
=
0
k
(
−
1
)
i
e
2
i
,
{\displaystyle \operatorname {tg} \left(\sum _{i=1}^{2k}\alpha _{i}\right)={\frac {e_{1}-e_{3}+e_{5}-\cdots +(-1)^{k+1}e_{2k-1}}{e_{0}-e_{2}+e_{4}-\cdots +(-1)^{k}e_{2k}}}={\frac {\sum _{i=1}^{k}(-1)^{i+1}e_{2i-1}}{\sum _{i=0}^{k}(-1)^{i}e_{2i}}},\!}
tg
(
∑
i
=
1
2
k
+
1
α
i
)
=
e
1
−
e
3
+
e
5
−
⋯
+
(
−
1
)
k
e
2
k
+
1
e
0
−
e
2
+
e
4
−
⋯
+
(
−
1
)
k
e
2
k
=
∑
i
=
1
k
(
−
1
)
i
e
2
i
+
1
∑
i
=
0
k
(
−
1
)
i
e
2
i
.
{\displaystyle \operatorname {tg} \left(\sum _{i=1}^{2k+1}\alpha _{i}\right)={\frac {e_{1}-e_{3}+e_{5}-\cdots +(-1)^{k}e_{2k+1}}{e_{0}-e_{2}+e_{4}-\cdots +(-1)^{k}e_{2k}}}={\frac {\sum _{i=1}^{k}(-1)^{i}e_{2i+1}}{\sum _{i=0}^{k}(-1)^{i}e_{2i}}}.\!}
Наприклад:
t
g
(
α
1
+
α
2
)
=
e
1
e
0
−
e
2
=
x
1
+
x
2
1
−
x
1
x
2
=
t
g
α
1
+
t
g
α
2
1
−
t
g
α
1
t
g
α
2
,
t
g
(
α
1
+
α
2
+
α
3
)
=
e
1
−
e
3
e
0
−
e
2
=
(
x
1
+
x
2
+
x
3
)
−
(
x
1
x
2
x
3
)
1
−
(
x
1
x
2
+
x
1
x
3
+
x
2
x
3
)
=
t
g
α
1
+
t
g
α
2
+
t
g
α
3
−
t
g
α
1
t
g
α
2
t
g
α
3
1
−
(
t
g
α
1
t
g
α
2
+
t
g
α
1
t
g
α
3
+
t
g
α
2
t
g
α
3
)
,
t
g
(
α
1
+
α
2
+
α
3
+
α
4
)
=
e
1
−
e
3
e
0
−
e
2
+
e
4
=
(
x
1
+
x
2
+
x
3
+
x
4
)
−
(
x
1
x
2
x
3
+
x
1
x
2
x
4
+
x
1
x
3
x
4
+
x
2
x
3
x
4
)
1
−
(
x
1
x
2
+
x
1
x
3
+
x
1
x
4
+
x
2
x
3
+
x
2
x
4
+
x
3
x
4
)
+
(
x
1
x
2
x
3
x
4
)
{\displaystyle {\begin{aligned}\mathrm {tg} (\alpha _{1}+\alpha _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\mathrm {tg} \,\alpha _{1}+\mathrm {tg} \,\alpha _{2}}{1\ -\ \mathrm {tg} \,\alpha _{1}\mathrm {tg} \,\alpha _{2}}},\\[8pt]\mathrm {tg} (\alpha _{1}+\alpha _{2}+\alpha _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}}={\frac {\mathrm {tg} \,\alpha _{1}+\mathrm {tg} \,\alpha _{2}+\mathrm {tg} \,\alpha _{3}-\mathrm {tg} \,\alpha _{1}\mathrm {tg} \,\alpha _{2}\mathrm {tg} \,\alpha _{3}}{1-(\mathrm {tg} \,\alpha _{1}\,\mathrm {tg} \,\alpha _{2}+\mathrm {tg} \,\alpha _{1}\,\mathrm {tg} \,\alpha _{3}+\mathrm {tg} \,\alpha _{2}\mathrm {tg} \,\alpha _{3})}},\\[8pt]\mathrm {tg} (\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}}\end{aligned}}}
і так далі.
ред.
sec
(
∑
i
n
α
i
)
=
∏
i
n
sec
α
i
e
0
−
e
2
+
e
4
−
⋯
=
∏
i
n
sec
α
i
∑
0
≤
2
k
≤
n
(
−
1
)
k
e
2
k
csc
(
∑
i
n
α
i
)
=
∏
i
n
sec
α
i
e
1
−
e
3
+
e
5
−
⋯
=
∏
i
n
sec
α
i
∑
1
≤
2
k
+
1
≤
n
(
−
1
)
k
e
2
k
+
1
{\displaystyle {\begin{aligned}\sec \left(\sum _{i}^{n}\alpha _{i}\right)&={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{\displaystyle \sum _{0\leq 2k\leq n}(-1)^{k}e_{2k}}}\\[8pt]\csc \left(\sum _{i}^{n}\alpha _{i}\right)&={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{\displaystyle \sum _{1\leq 2k+1\leq n}(-1)^{k}e_{2k+1}}}\end{aligned}}}
де e k елементарні симетричні многочлени степеня k від n змінних (дивись пункт тангенси від сум аргументів )
x
i
=
tg
α
i
i
=
1
,
2
,
…
,
n
.
{\displaystyle x_{i}=\operatorname {tg} \alpha _{i}\quad i=1,2,\ldots ,n.}
Наприклад,
sec
(
α
+
β
+
γ
)
=
sec
α
sec
β
sec
γ
1
−
t
g
α
t
g
β
−
t
g
α
t
g
γ
−
t
g
β
t
g
γ
,
csc
(
α
+
β
+
γ
)
=
sec
α
sec
β
sec
γ
t
g
α
+
t
g
β
+
t
g
γ
−
t
g
α
t
g
β
tg
γ
.
{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\mathrm {tg} \,\alpha \,\mathrm {tg} \,\beta -\mathrm {tg} \,\alpha \,\mathrm {tg} \,\gamma -\mathrm {tg} \,\beta \,\mathrm {tg} \,\gamma }},\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\mathrm {tg} \,\alpha +\mathrm {tg} \,\beta +\mathrm {tg} \,\gamma -\mathrm {tg} \,\alpha \,\mathrm {tg} \,\beta \operatorname {tg} \gamma }}.\end{aligned}}}
ред.
ред.
Формули потрійного кута
sin
3
α
=
3
sin
α
−
4
sin
3
α
=
4
sin
α
sin
(
π
3
−
α
)
sin
(
π
3
+
α
)
{\displaystyle \sin 3\alpha =3\sin \alpha -4\sin ^{3}\alpha =4\sin \alpha \sin \left({\frac {\pi }{3}}-\alpha \right)\sin \left({\frac {\pi }{3}}+\alpha \right)\,}
cos
3
α
=
4
cos
3
α
−
3
cos
α
=
4
cos
α
cos
(
π
3
−
α
)
cos
(
π
3
+
α
)
{\displaystyle \cos 3\alpha =4\cos ^{3}\alpha -3\cos \alpha =4\cos \alpha \cos \left({\frac {\pi }{3}}-\alpha \right)\cos \left({\frac {\pi }{3}}+\alpha \right)\,}
tg
3
α
=
3
tg
α
−
tg
3
α
1
−
3
tg
2
α
=
t
g
α
tg
(
π
3
−
α
)
tg
(
π
3
+
α
)
{\displaystyle \operatorname {tg} 3\alpha ={\frac {3\operatorname {tg} \alpha -\operatorname {tg} ^{3}\alpha }{1-3\operatorname {tg} ^{2}\alpha }}=\mathop {\mathrm {tg} } \,\alpha \operatorname {tg} \left({\frac {\pi }{3}}-\alpha \right)\operatorname {tg} \left({\frac {\pi }{3}}+\alpha \right)}
ctg
3
α
=
3
ctg
α
−
ctg
3
α
1
−
3
ctg
2
α
=
c
t
g
α
ctg
(
π
3
−
α
)
ctg
(
π
3
+
α
)
{\displaystyle \operatorname {ctg} 3\alpha ={\frac {3\operatorname {ctg} \alpha -\operatorname {ctg} ^{3}\alpha }{1-3\operatorname {ctg} ^{2}\alpha }}=\mathrm {ctg} \,\alpha \,\operatorname {ctg} \left({\frac {\pi }{3}}-\alpha \right)\operatorname {ctg} \left({\frac {\pi }{3}}+\alpha \right)}
ред.
Формули кратних кутів
sin
(
n
α
)
=
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
+
1
)
cos
n
−
2
k
−
1
α
sin
2
k
+
1
α
{\displaystyle \sin(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\cos ^{n-2k-1}\alpha \,\sin ^{2k+1}\alpha }
cos
(
n
α
)
=
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
)
cos
n
−
2
k
α
sin
2
k
α
{\displaystyle \cos(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\cos ^{n-2k}\alpha \,\sin ^{2k}\alpha }
t
g
(
n
α
)
=
sin
(
n
α
)
cos
(
n
α
)
=
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
+
1
)
tg
2
k
+
1
α
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
)
tg
2
k
α
{\displaystyle \mathrm {tg} (n\alpha )={\frac {\sin(n\alpha )}{\cos(n\alpha )}}={\dfrac {\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\operatorname {tg} ^{2k+1}\alpha }}{\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\operatorname {tg} ^{2k}\alpha }}}}
c
t
g
(
n
α
)
=
cos
(
n
α
)
sin
(
n
α
)
=
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
)
ctg
n
−
2
k
α
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
+
1
)
ctg
n
−
2
k
−
1
α
{\displaystyle \mathrm {ctg} (n\alpha )={\frac {\cos(n\alpha )}{\sin(n\alpha )}}={\dfrac {\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\operatorname {ctg} ^{n-2k}\alpha }}{\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\operatorname {ctg} ^{n-2k-1}\alpha }}}}
де
[
n
]
{\displaystyle [n]}
ціла частина числа
n
{\displaystyle n}
(
n
k
)
{\displaystyle {\binom {n}{k}}}
біноміальний коефіцієнт .
Вивід формул
Формули для кратних кутів виводяться за допомогою формули Муавра
(
cos
α
+
i
sin
α
)
n
=
cos
(
n
α
)
+
i
sin
(
n
α
)
,
i
2
=
−
1.
{\displaystyle \left(\cos \alpha +i\sin \alpha \right)^{n}=\cos \left(n\alpha \right)+i\sin \left(n\alpha \right),\quad i^{2}=-1.}
Розкриємо праву частину рівності за формулою бінома Ньютона
(
cos
α
+
i
sin
α
)
n
=
∑
q
=
0
n
(
n
q
)
(
cos
α
)
n
−
q
(
i
sin
α
)
q
.
{\displaystyle \left(\cos \alpha +i\sin \alpha \right)^{n}=\sum _{q=0}^{n}{n \choose q}(\cos \alpha )^{n-q}(i\sin \alpha )^{q}.}
Врахувавши, що
i
2
k
=
(
−
1
)
k
,
i
2
k
+
1
=
(
−
1
)
k
i
,
k
∈
Z
+
,
{\displaystyle i^{2k}=(-1)^{k},\,i^{2k+1}=(-1)^{k}i,\,k\in \mathbb {Z} _{+},}
(
cos
α
+
i
sin
α
)
n
=
∑
k
=
0
[
n
/
2
]
(
n
2
k
)
(
−
1
)
k
(
cos
α
)
n
−
2
k
(
sin
α
)
2
k
+
i
(
∑
k
=
0
[
n
/
2
]
(
n
2
k
+
1
)
(
−
1
)
k
(
cos
α
)
n
−
2
k
−
1
(
sin
α
)
2
k
+
1
)
.
{\displaystyle \left(\cos \alpha +i\sin \alpha \right)^{n}=\sum _{k=0}^{[n/2]}{n \choose 2k}(-1)^{k}(\cos \alpha )^{n-2k}(\sin \alpha )^{2k}+i\left(\sum _{k=0}^{[n/2]}{n \choose 2k+1}(-1)^{k}(\cos \alpha )^{n-2k-1}(\sin \alpha )^{2k+1}\right).}
Підставимо отриману рівність у формулу Муавра
∑
k
=
0
[
n
/
2
]
(
n
2
k
)
(
−
1
)
k
(
cos
α
)
n
−
2
k
(
sin
α
)
2
k
+
i
(
∑
k
=
0
[
n
/
2
]
(
n
2
k
+
1
)
(
−
1
)
k
(
cos
α
)
n
−
2
k
−
1
(
sin
α
)
2
k
+
1
)
=
cos
(
n
α
)
+
i
sin
(
n
α
)
.
{\displaystyle \sum _{k=0}^{[n/2]}{n \choose 2k}(-1)^{k}(\cos \alpha )^{n-2k}(\sin \alpha )^{2k}+i\left(\sum _{k=0}^{[n/2]}{n \choose 2k+1}(-1)^{k}(\cos \alpha )^{n-2k-1}(\sin \alpha )^{2k+1}\right)=\cos \left(n\alpha \right)+i\sin \left(n\alpha \right).}
Оскільки два комплексні числа рівні тоді і лише тоді коли рівні їхні дійсні та уявні частини, то з останньої рівності отримуємо шукані формули
sin
(
n
α
)
=
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
+
1
)
cos
n
−
2
k
−
1
α
sin
2
k
+
1
α
,
{\displaystyle \sin(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\cos ^{n-2k-1}\alpha \,\sin ^{2k+1}\alpha ,}
cos
(
n
α
)
=
∑
k
=
0
[
n
/
2
]
(
−
1
)
k
(
n
2
k
)
cos
n
−
2
k
α
sin
2
k
α
.
{\displaystyle \cos(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\cos ^{n-2k}\alpha \,\sin ^{2k}\alpha .}
sin
(
n
+
1
)
α
=
2
sin
α
cos
n
α
+
sin
(
n
−
1
)
α
,
{\displaystyle \sin(n+1)\alpha =2\sin \alpha \cos n\alpha +\sin(n-1)\alpha ,}
cos
(
n
+
1
)
α
=
2
cos
α
cos
n
α
+
cos
(
n
−
1
)
α
.
{\displaystyle \cos(n+1)\alpha =2\cos \alpha \cos n\alpha +\cos(n-1)\alpha .}
t
g
(
n
+
1
)
α
=
t
g
(
n
α
)
+
tg
α
1
−
t
g
(
n
α
)
tg
α
.
{\displaystyle \mathrm {tg} (n+1)\alpha ={\frac {\mathrm {tg} (n\alpha )+\operatorname {tg} \alpha }{1-\mathrm {tg} (n\alpha )\operatorname {tg} \alpha }}.}
c
t
g
(
n
+
1
)
α
=
c
t
g
(
n
α
)
ctg
α
−
1
c
t
g
(
n
α
)
+
ctg
α
.
{\displaystyle \mathrm {ctg} (n+1)\alpha ={\frac {\mathrm {ctg} (n\alpha )\operatorname {ctg} \alpha -1}{\mathrm {ctg} (n\alpha )+\operatorname {ctg} \alpha }}.}
ред.
Мають місце такі співвідношення:
cos
n
α
=
T
n
(
cos
α
)
,
sin
2
n
α
=
1
−
T
n
(
1
−
2
sin
2
α
)
2
,
{\displaystyle \cos n\alpha =T_{n}(\cos \alpha ),\quad \sin ^{2}n\alpha ={\frac {1-T_{n}(1-2\sin ^{2}\alpha )}{2}},}
де
T
n
(
x
)
{\displaystyle T_{n}(x)}
поліном Чебишова першого роду степеня n.
ред.
sin
2
m
α
=
2
m
sin
α
cos
α
∏
k
=
1
m
−
1
(
1
−
sin
2
α
sin
2
π
k
2
m
)
,
cos
2
m
α
=
∏
k
=
1
m
(
1
−
sin
2
α
sin
2
π
(
2
k
−
1
)
4
m
)
,
m
∈
N
,
{\displaystyle \sin 2m\alpha =2m\sin \alpha \cos \alpha \prod _{k=1}^{m-1}\left(1-{\frac {\sin ^{2}\alpha }{\displaystyle \sin ^{2}{\frac {\pi k}{2m}}}}\right),\quad \cos 2m\alpha =\prod _{k=1}^{m}\left(1-{\frac {\sin ^{2}\alpha }{\displaystyle \sin ^{2}{\frac {\pi (2k-1)}{4m}}}}\right),\quad m\in \mathbb {N} ,}
sin
(
2
m
−
1
)
α
=
(
2
m
−
1
)
sin
α
∏
k
=
1
m
−
1
(
1
−
sin
2
α
sin
2
π
k
2
m
−
1
)
,
cos
(
2
m
−
1
)
α
=
cos
α
∏
k
=
1
m
(
1
−
sin
2
α
sin
2
π
(
2
k
−
1
)
2
(
2
m
−
1
)
)
,
m
∈
N
,
{\displaystyle \sin(2m-1)\alpha =(2m-1)\sin \alpha \prod _{k=1}^{m-1}\left(1-{\frac {\sin ^{2}\alpha }{\displaystyle \sin ^{2}{\frac {\pi k}{2m-1}}}}\right),\quad \cos(2m-1)\alpha =\cos \alpha \prod _{k=1}^{m}\left(1-{\frac {\sin ^{2}\alpha }{\displaystyle \sin ^{2}{\frac {\pi (2k-1)}{2(2m-1)}}}}\right),\quad m\in \mathbb {N} ,}
sin
n
α
=
2
n
−
1
∏
k
=
0
n
−
1
sin
(
α
+
k
π
n
)
,
cos
n
α
=
2
n
−
1
∏
k
=
1
n
sin
(
α
+
(
2
k
−
1
)
π
2
n
)
.
{\displaystyle \sin n\alpha =2^{n-1}\prod _{k=0}^{n-1}\sin \left(\alpha +{\frac {k\pi }{n}}\right),\quad \cos n\alpha =2^{n-1}\prod _{k=1}^{n}\sin \left(\alpha +{\frac {(2k-1)\pi }{2n}}\right).}
ред.
ред.
ред.
ред.
Формули перетворення добутків функцій
sin
α
sin
β
=
cos
(
α
−
β
)
−
cos
(
α
+
β
)
2
{\displaystyle \sin \alpha \sin \beta ={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{2}}}
(28)
sin
α
cos
β
=
sin
(
α
+
β
)
+
sin
(
α
−
β
)
2
{\displaystyle \sin \alpha \cos \beta ={\frac {\sin(\alpha +\beta )+\sin(\alpha -\beta )}{2}}}
(29)
cos
α
cos
β
=
cos
(
α
+
β
)
+
cos
(
α
−
β
)
2
{\displaystyle \cos \alpha \cos \beta ={\frac {\cos(\alpha +\beta )+\cos(\alpha -\beta )}{2}}}
(30)
tg
α
tg
β
=
cos
(
α
−
β
)
−
cos
(
α
+
β
)
cos
(
α
+
β
)
+
cos
(
α
−
β
)
{\displaystyle \operatorname {tg} \alpha \operatorname {tg} \beta ={\frac {\cos(\alpha -\beta )-\cos(\alpha +\beta )}{\cos(\alpha +\beta )+\cos(\alpha -\beta )}}}
cos
α
cos
β
cos
γ
=
cos
(
α
+
β
+
γ
)
+
cos
(
α
−
β
+
γ
)
+
cos
(
α
+
β
−
γ
)
+
cos
(
β
+
γ
−
α
)
4
{\displaystyle \cos \alpha \cos \beta \cos \gamma ={\frac {\cos(\alpha +\beta +\gamma )+\cos(\alpha -\beta +\gamma )+\cos(\alpha +\beta -\gamma )+\cos(\beta +\gamma -\alpha )}{4}}}
(31)
sin
α
cos
β
cos
γ
=
sin
(
α
+
β
+
γ
)
+
sin
(
α
−
β
+
γ
)
+
sin
(
α
+
β
−
γ
)
−
sin
(
β
+
γ
−
α
)
4
{\displaystyle \sin \alpha \cos \beta \cos \gamma ={\frac {\sin(\alpha +\beta +\gamma )+\sin(\alpha -\beta +\gamma )+\sin(\alpha +\beta -\gamma )-\sin(\beta +\gamma -\alpha )}{4}}}
(32)
sin
α
sin
β
cos
γ
=
−
cos
(
α
+
β
+
γ
)
+
cos
(
α
−
β
+
γ
)
−
cos
(
α
+
β
−
γ
)
−
cos
(
β
+
γ
−
α
)
4
{\displaystyle \,\sin \alpha \sin \beta \cos \gamma ={\frac {-\cos(\alpha +\beta +\gamma )+\cos(\alpha -\beta +\gamma )-\cos(\alpha +\beta -\gamma )-\cos(\beta +\gamma -\alpha )}{4}}}
(33)
sin
α
sin
β
sin
γ
=
−
sin
(
α
+
β
+
γ
)
+
sin
(
α
−
β
+
γ
)
+
sin
(
α
+
β
−
γ
)
+
sin
(
β
+
γ
−
α
)
4
{\displaystyle \sin \alpha \sin \beta \sin \gamma ={\frac {-\sin(\alpha +\beta +\gamma )+\sin(\alpha -\beta +\gamma )+\sin(\alpha +\beta -\gamma )+\sin(\beta +\gamma -\alpha )}{4}}}
(34)
ред.
Формули перетворення суми функцій
sin
α
±
sin
β
=
2
sin
α
±
β
2
cos
α
∓
β
2
{\displaystyle \sin \alpha \pm \sin \beta =2\sin {\frac {\alpha \pm \beta }{2}}\cos {\frac {\alpha \mp \beta }{2}}}
(35)
cos
α
+
cos
β
=
2
cos
α
+
β
2
cos
α
−
β
2
{\displaystyle \cos \alpha +\cos \beta =2\cos {\frac {\alpha +\beta }{2}}\cos {\frac {\alpha -\beta }{2}}}
(36)
cos
α
−
cos
β
=
−
2
sin
α
+
β
2
sin
α
−
β
2
{\displaystyle \cos \alpha -\cos \beta =-2\sin {\frac {\alpha +\beta }{2}}\sin {\frac {\alpha -\beta }{2}}}
(37)
t
g
α
±
t
g
β
=
sin
(
α
±
β
)
cos
α
cos
β
{\displaystyle \mathop {\mathrm {tg} } \alpha \pm \mathop {\mathrm {tg} } \beta ={\frac {\sin(\alpha \pm \beta )}{\cos \alpha \cos \beta }}}
(38)
c
t
g
α
±
c
t
g
β
=
sin
(
β
±
α
)
sin
α
sin
β
{\displaystyle \mathop {\mathrm {ctg} } \alpha \pm \mathop {\mathrm {ctg} } \beta ={\frac {\sin(\beta \pm \alpha )}{\sin \alpha \sin \beta }}}
(39)
t
g
α
±
c
t
g
β
=
±
cos
(
α
∓
β
)
cos
α
sin
β
{\displaystyle \mathop {\mathrm {tg} } \alpha \pm \mathop {\mathrm {ctg} } \beta ={\frac {\pm \cos(\alpha \mp \beta )}{\cos \alpha \sin \beta }}}
(40)
t
g
α
+
c
t
g
α
=
1
cos
α
sin
α
=
2
csc
2
α
{\displaystyle \mathop {\mathrm {tg} } \alpha +\mathop {\mathrm {ctg} } \alpha ={\frac {1}{\cos \alpha \sin \alpha }}=2\csc 2\alpha }
(41)
t
g
α
−
c
t
g
α
=
−
2
c
t
g
2
α
{\displaystyle \mathop {\mathrm {tg} } \alpha -\mathop {\mathrm {ctg} } \alpha =-2\mathop {\mathrm {ctg} } 2\alpha }
(42)
cos
α
+
sin
α
=
2
cos
(
π
4
−
α
)
=
2
sin
(
π
4
+
α
)
{\displaystyle \cos \alpha +\sin \alpha ={\sqrt {2}}\cos \left({\frac {\pi }{4}}-\alpha \right)={\sqrt {2}}\sin \left({\frac {\pi }{4}}+\alpha \right)}
(43)
cos
α
−
sin
α
=
2
sin
(
π
4
−
α
)
=
2
cos
(
π
4
+
α
)
{\displaystyle \cos \alpha -\sin \alpha ={\sqrt {2}}\sin \left({\frac {\pi }{4}}-\alpha \right)={\sqrt {2}}\cos \left({\frac {\pi }{4}}+\alpha \right)}
(43)
∑
k
=
1
n
cos
(
2
k
−
1
)
α
=
cos
α
+
cos
3
α
+
cos
5
α
+
…
+
cos
(
2
n
−
1
)
α
=
sin
2
n
α
2
sin
α
,
n
≥
1
,
{\displaystyle \sum _{k=1}^{n}\cos(2k-1)\alpha =\cos \alpha +\cos 3\alpha +\cos 5\alpha +\ldots +\cos(2n-1)\alpha ={\frac {\sin 2n\alpha }{2\sin \alpha }},\quad n\geq 1,}
{\displaystyle }
∑
k
=
0
n
sin
(
2
k
−
1
)
α
=
sin
α
+
sin
3
α
+
sin
5
α
+
…
+
sin
(
2
n
−
1
)
α
=
sin
2
n
α
sin
α
,
n
≥
1
,
{\displaystyle \sum _{k=0}^{n}\sin(2k-1)\alpha =\sin \alpha +\sin 3\alpha +\sin 5\alpha +\ldots +\sin(2n-1)\alpha ={\frac {\sin ^{2}n\alpha }{\sin \alpha }},\quad n\geq 1,}
{\displaystyle }
∑
k
=
0
n
cos
(
φ
+
k
α
)
=
cos
φ
+
cos
(
φ
+
α
)
+
cos
(
φ
+
2
α
)
+
…
+
cos
(
φ
+
n
α
)
=
cos
(
φ
+
n
α
2
)
sin
(
n
+
1
)
α
2
sin
α
2
,
α
≠
0
,
n
≥
1
,
{\displaystyle \sum _{k=0}^{n}\cos(\varphi +k\alpha )=\cos \varphi +\cos(\varphi +\alpha )+\cos(\varphi +2\alpha )+\ldots +\cos(\varphi +n\alpha )={\frac {\displaystyle \cos \left(\varphi +{\frac {n\alpha }{2}}\right)\sin {\frac {(n+1)\alpha }{2}}}{\displaystyle \sin {\frac {\alpha }{2}}}},\quad \alpha \neq 0,n\geq 1,}
∑
k
=
0
n
sin
(
φ
+
k
α
)
=
sin
φ
+
sin
(
φ
+
α
)
+
sin
(
φ
+
2
α
)
+
…
+
sin
(
φ
+
n
α
)
=
sin
(
φ
+
n
α
2
)
sin
(
n
+
1
)
α
2
sin
α
2
,
α
≠
0
,
n
≥
1.
{\displaystyle \sum _{k=0}^{n}\sin(\varphi +k\alpha )=\sin \varphi +\sin(\varphi +\alpha )+\sin(\varphi +2\alpha )+\ldots +\sin(\varphi +n\alpha )={\frac {\displaystyle \sin \left(\varphi +{\frac {n\alpha }{2}}\right)\sin {\frac {(n+1)\alpha }{2}}}{\displaystyle \sin {\frac {\alpha }{2}}}},\quad \alpha \neq 0,n\geq 1.}
∑
k
=
0
n
cos
2
(
k
α
)
=
3
+
2
n
+
csc
α
sin
(
2
n
+
1
)
α
4
,
∑
k
=
0
n
sin
2
(
k
α
)
=
1
+
2
n
−
csc
α
sin
(
2
n
+
1
)
α
4
,
n
≥
1
,
{\displaystyle \sum _{k=0}^{n}\cos ^{2}(k\alpha )={\frac {\displaystyle 3+2n+\csc \alpha \sin(2n+1)\alpha }{4}},\quad \sum _{k=0}^{n}\sin ^{2}(k\alpha )={\frac {\displaystyle 1+2n-\csc \alpha \sin(2n+1)\alpha }{4}},\quad n\geq 1,}
∑
k
=
1
n
−
1
k
cos
(
k
α
)
=
n
sin
2
n
−
1
2
α
2
sin
α
2
−
1
−
cos
n
α
4
sin
2
α
2
,
∑
k
=
1
n
−
1
k
sin
(
k
α
)
=
sin
n
α
4
sin
2
α
2
−
n
cos
2
n
−
1
2
α
2
sin
α
2
,
n
≥
2
,
{\displaystyle \sum _{k=1}^{n-1}k\cos(k\alpha )={\frac {n\displaystyle \sin {\frac {2n-1}{2}}\alpha }{2\displaystyle \sin {\frac {\alpha }{2}}}}-{\frac {1-\cos n\alpha }{4\displaystyle \sin ^{2}{\frac {\alpha }{2}}}},\quad \sum _{k=1}^{n-1}k\sin(k\alpha )={\frac {\sin n\alpha }{4\displaystyle \sin ^{2}{\frac {\alpha }{2}}}}-{\frac {n\displaystyle \cos {\frac {2n-1}{2}}\alpha }{2\displaystyle \sin {\frac {\alpha }{2}}}},\quad n\geq 2,}
∑
k
=
0
n
p
k
cos
(
k
α
)
=
1
−
p
cos
α
−
p
n
+
1
cos
(
n
+
1
)
α
−
p
n
+
2
cos
n
α
1
−
2
p
cos
α
+
p
2
,
n
≥
1
,
{\displaystyle \sum _{k=0}^{n}p^{k}\cos(k\alpha )={\frac {1-p\cos \alpha -p^{n+1}\cos(n+1)\alpha -p^{n+2}\cos n\alpha }{1-2p\cos \alpha +p^{2}}},\quad n\geq 1,}
∑
k
=
0
n
p
k
sin
(
k
α
)
=
p
sin
α
−
p
n
+
1
sin
(
n
+
1
)
α
−
p
n
+
2
sin
n
α
1
−
2
p
cos
α
+
p
2
,
n
≥
1.
{\displaystyle \sum _{k=0}^{n}p^{k}\sin(k\alpha )={\frac {p\sin \alpha -p^{n+1}\sin(n+1)\alpha -p^{n+2}\sin n\alpha }{1-2p\cos \alpha +p^{2}}},\quad n\geq 1.}
p
{\displaystyle p}
|
p
|
<
1
{\displaystyle |p|<1}
n
→
∞
{\displaystyle n\rightarrow \infty }
∑
k
=
0
∞
p
k
cos
(
k
α
)
=
1
−
p
cos
α
1
−
2
p
cos
α
+
p
2
,
∑
k
=
0
∞
p
k
sin
(
k
α
)
=
p
sin
α
1
−
2
p
cos
α
+
p
2
.
{\displaystyle \sum _{k=0}^{\infty }p^{k}\cos(k\alpha )={\frac {1-p\cos \alpha }{1-2p\cos \alpha +p^{2}}},\quad \sum _{k=0}^{\infty }p^{k}\sin(k\alpha )={\frac {p\sin \alpha }{1-2p\cos \alpha +p^{2}}}.}
ред.
Сума виду
D
n
(
x
)
=
1
2
+
∑
k
=
1
n
cos
(
k
x
)
=
sin
(
(
n
+
1
2
)
x
)
2
sin
(
x
2
)
.
{\displaystyle D_{n}(x)={\frac {1}{2}}+\sum _{k=1}^{n}\cos(kx)={\frac {\sin \left(\left(n+\displaystyle {\frac {1}{2}}\right)x\right)}{2\sin \left(\displaystyle {\frac {x}{2}}\right)}}.}
називається ядром Діріхле .
А функція
Φ
n
(
x
)
=
1
n
+
1
∑
k
=
0
n
D
k
(
x
)
,
{\displaystyle \Phi _{n}(x)={\frac {1}{n+1}}\sum _{k=0}^{n}D_{k}(x),}
називається ядром Феєра
Φ
n
(
x
)
=
1
2
(
n
+
1
)
(
sin
n
+
1
2
x
sin
x
2
)
2
=
1
2
(
n
+
1
)
1
−
cos
(
(
n
+
1
)
x
)
1
−
cos
x
{\displaystyle \Phi _{n}(x)={\frac {1}{2(n+1)}}\left({\frac {\sin \displaystyle {\frac {n+1}{2}}x}{\sin \displaystyle {\frac {x}{2}}}}\right)^{2}={\frac {1}{2(n+1)}}{\frac {1-\cos((n+1)x)}{1-\cos x}}}
Вони використані при сумуванні рядів Фур'є .
ред.
sin
x
=
x
∏
n
=
1
∞
(
1
−
x
2
π
2
n
2
)
,
cos
x
=
∏
n
=
1
∞
(
1
−
x
2
π
2
(
n
−
1
2
)
2
)
,
{\displaystyle \sin x=x\prod _{n=1}^{\infty }\left(1-{\frac {x^{2}}{\pi ^{2}n^{2}}}\right),\qquad \cos x=\prod _{n=1}^{\infty }\left(1-\displaystyle {\frac {x^{2}}{\pi ^{2}\left(n-\displaystyle {\frac {1}{2}}\right)^{2}}}\right),}
ред.
α
+
β
+
γ
=
π
,
{\displaystyle \alpha +\beta +\gamma =\pi ,}
тоді
sin
2
α
+
sin
2
β
+
sin
2
γ
=
4
sin
α
sin
β
sin
γ
,
{\displaystyle \sin 2\alpha +\sin 2\beta +\sin 2\gamma =4\sin \alpha \sin \beta \sin \gamma ,\,}
sin
α
+
sin
β
+
sin
γ
=
4
cos
α
2
cos
β
2
cos
γ
2
,
{\displaystyle \sin \alpha +\sin \beta +\sin \gamma =4\cos {\frac {\alpha }{2}}\cos {\frac {\beta }{2}}\cos {\frac {\gamma }{2}},\,}
sin
2
α
+
sin
2
β
+
sin
2
γ
=
2
cos
α
cos
β
cos
γ
+
2
,
{\displaystyle \sin ^{2}\alpha +\sin ^{2}\beta +\sin ^{2}\gamma =2\cos \alpha \cos \beta \cos \gamma +2,\,}
cos
2
α
+
cos
2
β
+
cos
2
γ
=
−
4
cos
α
cos
β
cos
γ
−
1
,
{\displaystyle \cos 2\alpha +\cos 2\beta +\cos 2\gamma =-4\cos \alpha \cos \beta \cos \gamma -1,\,}
cos
α
+
cos
β
+
cos
γ
=
4
sin
α
2
sin
β
2
sin
γ
2
+
1
,
{\displaystyle \cos \alpha +\cos \beta +\cos \gamma =4\sin {\frac {\alpha }{2}}\sin {\frac {\beta }{2}}\sin {\frac {\gamma }{2}}+1,}
cos
2
α
+
cos
2
β
+
cos
2
γ
=
−
2
cos
α
cos
β
cos
γ
+
1
,
{\displaystyle \cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma =-2\cos \alpha \cos \beta \cos \gamma +1,\,}
t
g
α
+
t
g
β
+
t
g
γ
=
t
g
α
t
g
β
t
g
γ
,
{\displaystyle \mathrm {tg} \,\alpha +\mathrm {tg} \,\beta +\mathrm {tg} \,\gamma =\mathrm {tg} \,\alpha \,\mathrm {tg} \,\beta \,\mathrm {tg} \,\gamma ,\,}
t
g
β
2
t
g
γ
2
+
t
g
γ
2
t
g
α
2
+
t
g
α
2
t
g
β
2
=
1
,
{\displaystyle \mathrm {tg} \,{\frac {\beta }{2}}\mathrm {tg} \,{\frac {\gamma }{2}}+\mathrm {tg} \,{\frac {\gamma }{2}}\mathrm {tg} \,{\frac {\alpha }{2}}+\mathrm {tg} \,{\frac {\alpha }{2}}\mathrm {tg} \,{\frac {\beta }{2}}=1,}
c
t
g
α
2
+
c
t
g
β
2
+
c
t
g
γ
2
=
c
t
g
α
2
⋅
c
t
g
β
2
⋅
c
t
g
γ
2
,
{\displaystyle \mathrm {ctg} \,{\frac {\alpha }{2}}+\mathrm {ctg} \,{\frac {\beta }{2}}+\mathrm {ctg} \,{\frac {\gamma }{2}}=\mathrm {ctg} \,{\frac {\alpha }{2}}\cdot \mathrm {ctg} \,{\frac {\beta }{2}}\cdot \mathrm {ctg} \,{\frac {\gamma }{2}},}
c
t
g
α
c
t
g
β
+
c
t
g
β
c
t
g
γ
+
c
t
g
γ
c
t
g
α
=
1.
{\displaystyle \mathrm {ctg} \,\alpha \,\mathrm {ctg} \,\beta +\mathrm {ctg} \,\beta \,\mathrm {ctg} \,\gamma +\mathrm {ctg} \,\gamma \,\mathrm {ctg} \,\alpha =1.\,}
Зауважимо, що наведені вище співвідношення справджуються, якщо
α
,
β
,
γ
{\displaystyle \alpha ,\beta ,\gamma }
α
+
β
+
γ
=
π
2
,
{\displaystyle \alpha +\beta +\gamma ={\frac {\pi }{2}},}
тоді
c
t
g
(
α
)
+
c
t
g
(
β
)
+
c
t
g
(
γ
)
=
c
t
g
(
α
)
c
t
g
(
β
)
c
t
g
(
γ
)
.
{\displaystyle \mathrm {ctg} (\alpha )+\mathrm {ctg} (\beta )+\mathrm {ctg} (\gamma )=\mathrm {ctg} (\alpha )\,\mathrm {ctg} (\beta )\,\mathrm {ctg} (\gamma ).}
α
+
β
+
γ
+
θ
=
π
,
{\displaystyle \alpha +\beta +\gamma +\theta =\pi ,}
тоді
sin
(
θ
+
α
)
sin
(
α
+
β
)
=
sin
(
α
+
β
)
sin
(
β
+
γ
)
=
sin
(
β
+
γ
)
sin
(
γ
+
θ
)
=
sin
(
γ
+
θ
)
sin
(
θ
+
α
)
=
sin
(
θ
)
sin
(
β
)
+
sin
(
α
)
sin
(
γ
)
.
{\displaystyle {\begin{aligned}\sin(\theta +\alpha )\sin(\alpha +\beta )&=\sin(\alpha +\beta )\sin(\beta +\gamma )\\&=\sin(\beta +\gamma )\sin(\gamma +\theta )\\&=\sin(\gamma +\theta )\sin(\theta +\alpha )\\&=\sin(\theta )\sin(\beta )+\sin(\alpha )\sin(\gamma ).\end{aligned}}}
ред.
arcsin
(
−
x
)
=
−
arcsin
(
x
)
,
arccos
(
−
x
)
=
π
−
arccos
(
x
)
,
arcsin
(
x
)
+
arccos
(
x
)
=
π
/
2
{\displaystyle \arcsin(-x)=-\arcsin(x),\quad \arccos(-x)=\pi -\arccos(x),\quad \arcsin(x)+\arccos(x)=\pi /2\;}
a
r
c
t
g
(
−
x
)
=
−
a
r
c
t
g
(
x
)
,
a
r
c
c
t
g
(
−
x
)
=
π
−
a
r
c
c
t
g
(
x
)
,
a
r
c
t
g
(
x
)
+
a
r
c
c
t
g
(
x
)
=
π
/
2.
{\displaystyle \mathrm {arctg} (-x)=-\mathrm {arctg} (x),\quad \mathrm {arcctg} (-x)=\pi -\mathrm {arcctg} (x),\quad \mathrm {arctg} (x)+\mathrm {arcctg} (x)=\pi /2.\;}
a
r
c
t
g
(
x
)
+
a
r
c
t
g
(
1
/
x
)
=
{
π
/
2
,
x
>
0
,
−
π
/
2
,
x
<
0.
{\displaystyle \mathrm {arctg} (x)+\mathrm {arctg} (1/x)=\left\{{\begin{matrix}\pi /2,&x>0,\\-\pi /2,&x<0.\end{matrix}}\right.}
Зв'язок між оберненими тригонометричними функціями для x>0
arccos
{\displaystyle \arccos }
arcsin
{\displaystyle \arcsin }
a
r
c
t
g
{\displaystyle \mathrm {arctg} \,}
a
r
c
c
t
g
{\displaystyle \mathrm {arcctg} \,}
arccos
x
=
{\displaystyle \arccos x=}
arccos
x
{\displaystyle \arccos x}
π
2
−
arcsin
x
{\displaystyle {\frac {\pi }{2}}-\arcsin x}
a
r
c
t
g
1
−
x
2
x
{\displaystyle \mathrm {arctg} \,{\frac {\sqrt {1-x^{2}}}{x}}}
a
r
c
c
t
g
x
1
−
x
2
{\displaystyle \mathrm {arcctg} \,{\frac {x}{\sqrt {1-x^{2}}}}}
arcsin
x
=
{\displaystyle \arcsin x=}
π
2
−
arccos
x
{\displaystyle {\frac {\pi }{2}}-\arccos x}
arcsin
x
{\displaystyle \arcsin x}
a
r
c
t
g
x
1
−
x
2
{\displaystyle \mathrm {arctg} \,{\frac {x}{\sqrt {1-x^{2}}}}}
a
r
c
c
t
g
1
−
x
2
x
{\displaystyle \mathrm {arcctg} \,{\frac {\sqrt {1-x^{2}}}{x}}}
a
r
c
t
g
x
=
{\displaystyle \mathrm {arctg} \,x=}
arccos
1
1
+
x
2
{\displaystyle \arccos {\frac {1}{\sqrt {1+x^{2}}}}}
arcsin
x
1
+
x
2
{\displaystyle \arcsin {\frac {x}{\sqrt {1+x^{2}}}}}
a
r
c
t
g
x
{\displaystyle \mathrm {arctg} \,x}
a
r
c
c
t
g
1
x
{\displaystyle \mathrm {arcctg} \,{\frac {1}{x}}}
a
r
c
c
t
g
x
=
{\displaystyle \mathrm {arcctg} \,x=}
arccos
x
1
+
x
2
{\displaystyle \arccos {\frac {x}{\sqrt {1+x^{2}}}}}
arcsin
1
1
+
x
2
{\displaystyle \arcsin {\frac {1}{\sqrt {1+x^{2}}}}}
a
r
c
t
g
1
x
{\displaystyle \mathrm {arctg} \,{\frac {1}{x}}}
a
r
c
c
t
g
x
{\displaystyle \mathrm {arcctg} \,x}
ред.
Поєднання тригонометричних та обернених їм функцій
sin
[
arccos
(
x
)
]
=
1
−
x
2
{\displaystyle \sin[\arccos(x)]={\sqrt {1-x^{2}}}\,}
t
g
[
arcsin
(
x
)
]
=
x
1
−
x
2
{\displaystyle \mathrm {tg} [\arcsin(x)]={\frac {x}{\sqrt {1-x^{2}}}}}
sin
[
a
r
c
t
g
(
x
)
]
=
x
1
+
x
2
{\displaystyle \sin[\mathrm {arctg} (x)]={\frac {x}{\sqrt {1+x^{2}}}}}
t
g
[
arccos
(
x
)
]
=
1
−
x
2
x
{\displaystyle \mathrm {tg} [\arccos(x)]={\frac {\sqrt {1-x^{2}}}{x}}}
cos
[
a
r
c
t
g
(
x
)
]
=
1
1
+
x
2
{\displaystyle \cos[\mathrm {arctg} (x)]={\frac {1}{\sqrt {1+x^{2}}}}}
c
t
g
[
arcsin
(
x
)
]
=
1
−
x
2
x
{\displaystyle \mathrm {ctg} [\arcsin(x)]={\frac {\sqrt {1-x^{2}}}{x}}}
cos
[
arcsin
(
x
)
]
=
1
−
x
2
{\displaystyle \cos[\arcsin(x)]={\sqrt {1-x^{2}}}\,}
c
t
g
[
arccos
(
x
)
]
=
x
1
−
x
2
{\displaystyle \mathrm {ctg} [\arccos(x)]={\frac {x}{\sqrt {1-x^{2}}}}}
ред.
Нехай
x
,
y
{\displaystyle x,y}
|
x
|
≤
1
,
|
y
|
≤
1
{\displaystyle |x|\leq 1,|y|\leq 1}
arcsin
(
x
)
+
arcsin
(
y
)
=
(
−
1
)
ε
arcsin
(
x
1
−
y
2
+
y
1
−
x
2
)
+
ε
π
,
ε
=
{
0
,
x
y
≤
0
,
sgn
x
,
x
y
>
0
,
{\displaystyle \arcsin(x)+\arcsin(y)=(-1)^{\varepsilon }\arcsin \left(x{\sqrt {1-y^{2}}}+y{\sqrt {1-x^{2}}}\right)+\varepsilon \pi ,\quad \varepsilon =\left\{{\begin{matrix}0,&xy\leq 0,\\\operatorname {sgn} x,&xy>0,\end{matrix}}\right.}
{\displaystyle \quad }
arccos
(
x
)
+
arccos
(
y
)
=
(
−
1
)
ε
arccos
(
x
y
−
1
−
y
2
1
−
x
2
)
+
ε
π
,
ε
=
{
0
,
x
+
y
≥
0
,
1
,
x
+
y
<
0
,
{\displaystyle \arccos(x)+\arccos(y)=(-1)^{\varepsilon }\arccos \left(xy-{\sqrt {1-y^{2}}}{\sqrt {1-x^{2}}}\right)+\varepsilon \pi ,\quad \varepsilon =\left\{{\begin{matrix}0,&x+y\geq 0,\\1,&x+y<0,\end{matrix}}\right.}
{\displaystyle \quad }
a
r
c
t
g
(
x
)
+
a
r
c
t
g
(
y
)
=
arctg
(
x
+
y
1
−
x
y
)
+
ε
π
,
ε
=
{
−
1
,
x
y
<
1
,
0
,
x
y
=
1
,
1
,
x
y
>
1.
{\displaystyle \mathrm {arctg} (x)+\mathrm {arctg} (y)=\operatorname {arctg} \left({\frac {x+y}{1-xy}}\right)+\varepsilon \pi ,\quad \varepsilon =\left\{{\begin{array}{ll}-1,&xy<1,\\0,&xy=1,\\1,&xy>1.\end{array}}\right.}
ред.
sin
x
=
a
{\displaystyle \operatorname {sin} x=a}
Якщо
|
a
|
>
1
{\displaystyle |a|>1}
Якщо
|
a
|
≤
1
{\displaystyle |a|\leq 1}
x
=
(
−
1
)
n
arcsin
a
+
π
n
;
n
∈
Z
{\displaystyle x=(-1)^{n}\arcsin a+\pi n;n\in \mathbb {Z} }
cos
x
=
a
{\displaystyle \operatorname {cos} x=a}
Якщо
|
a
|
>
1
{\displaystyle |a|>1}
Якщо
|
a
|
≤
1
{\displaystyle |a|\leq 1}
x
=
±
arccos
a
+
2
π
n
;
n
∈
Z
{\displaystyle x=\pm \arccos a+2\pi n;n\in \mathbb {Z} }
tg
x
=
a
{\displaystyle \operatorname {tg} x=a}
Розв'язком є число виду
x
=
arctg
a
+
π
n
;
n
∈
Z
{\displaystyle x=\operatorname {arctg} a+\pi n;n\in \mathbb {Z} }
ctg
x
=
a
{\displaystyle \operatorname {ctg} x=a}
Розв'язком є число виду
x
=
arcctg
a
+
π
n
;
n
∈
Z
{\displaystyle x=\operatorname {arcctg} a+\pi n;n\in \mathbb {Z} }
ред.
Вид нерівності
Множина розв'язків,
n
∈
Z
{\displaystyle n\in \mathbb {Z} }
sin
x
>
a
(
|
a
|
⩽
1
)
{\displaystyle \sin x>a\quad (|a|\leqslant 1)}
x
∈
(
arcsin
a
+
2
π
n
,
π
−
arcsin
a
+
2
π
n
)
{\displaystyle x\in (\arcsin a+2\pi n,\,\pi -\arcsin a+2\pi n)}
sin
x
<
a
(
|
a
|
⩽
1
)
{\displaystyle \sin x<a\quad (|a|\leqslant 1)}
x
∈
(
−
π
−
arcsin
a
+
2
π
n
,
arcsin
a
+
2
π
n
)
{\displaystyle x\in (-\pi -\arcsin a+2\pi n,\,\arcsin a+2\pi n)}
cos
x
>
a
(
|
a
|
⩽
1
)
{\displaystyle \cos x>a\quad (|a|\leqslant 1)}
x
∈
(
−
arccos
a
+
2
π
n
,
arccos
a
+
2
π
n
)
{\displaystyle x\in (-\arccos a+2\pi n,\,\arccos a+2\pi n)}
cos
x
<
a
(
|
a
|
⩽
1
)
{\displaystyle \cos x<a\quad (|a|\leqslant 1)}
x
∈
(
arccos
a
+
2
π
n
,
2
π
−
arccos
a
+
2
π
n
)
{\displaystyle x\in (\arccos a+2\pi n,\,2\pi -\arccos a+2\pi n)}
t
g
x
>
a
{\displaystyle \mathrm {tg} \,x>a}
x
∈
(
a
r
c
t
g
a
+
π
n
,
π
2
+
π
n
)
{\displaystyle x\in \left(\mathrm {arctg} \,a+\pi n,\,{\frac {\pi }{2}}+\pi n\right)}
t
g
x
<
a
{\displaystyle \mathrm {tg} \,x<a}
x
∈
(
−
π
2
+
π
n
,
a
r
c
t
g
a
+
π
n
)
{\displaystyle x\in \left(-{\frac {\pi }{2}}+\pi n,\,\mathrm {arctg} \,a+\pi n\right)}
c
t
g
x
>
a
{\displaystyle \mathrm {ctg} \,x>a}
x
∈
(
π
n
,
a
r
c
c
t
g
a
+
π
n
)
{\displaystyle x\in (\pi n,\,\mathrm {arcctg} \,a+\pi n)}
c
t
g
x
<
a
{\displaystyle \mathrm {ctg} \,x<a}
x
∈
(
a
r
c
t
g
a
+
π
n
,
π
n
)
{\displaystyle x\in (\mathrm {arctg} \,a+\pi n,\,\pi n)}
ред.
Для довільного
x
{\displaystyle x}
[
−
π
/
2
,
π
/
2
]
{\displaystyle [-\pi /2,\pi /2]}
2
π
|
x
|
⩽
|
sin
(
x
)
|
⩽
|
x
|
.
{\displaystyle {\frac {2}{\pi }}|x|\leqslant |{\sin(x)}|\leqslant |x|.}
ред.
Тотожності мають зміст лише тоді, коли існують обидві частини (тобто при
α
≠
π
+
2
π
n
{\displaystyle \alpha \neq \pi +2\pi n}
sin
α
=
2
tg
α
2
1
+
tg
2
α
2
,
cos
α
=
1
−
tg
2
α
2
1
+
tg
2
α
2
,
sec
α
=
1
+
tg
2
α
2
1
−
tg
2
α
2
;
{\displaystyle \operatorname {sin} \alpha ={\frac {2\operatorname {tg} \displaystyle {\frac {\alpha }{2}}}{1+\operatorname {tg} ^{2}\displaystyle {\frac {\alpha }{2}}}},\qquad \operatorname {cos} \alpha ={\frac {1-\operatorname {tg} ^{2}\displaystyle {\frac {\alpha }{2}}}{1+\operatorname {tg} ^{2}\displaystyle {\frac {\alpha }{2}}}},\qquad \sec \alpha ={\frac {1+\displaystyle \operatorname {tg} ^{2}{\frac {\alpha }{2}}}{1-\displaystyle \operatorname {tg} ^{2}{\frac {\alpha }{2}}}};}
tg
α
=
2
tg
α
2
1
−
tg
2
α
2
,
ctg
α
=
1
−
tg
2
α
2
2
tg
α
2
,
csc
α
=
1
+
tg
2
α
2
2
tg
α
2
.
{\displaystyle \operatorname {tg} \alpha ={\frac {2\operatorname {tg} \displaystyle {\frac {\alpha }{2}}}{1-\operatorname {tg} ^{2}\displaystyle {\frac {\alpha }{2}}}},\qquad \operatorname {ctg} \alpha ={\frac {1-\operatorname {tg} ^{2}\displaystyle {\frac {\alpha }{2}}}{2\operatorname {tg} \displaystyle {\frac {\alpha }{2}}}},\qquad \csc \alpha ={\frac {1+\displaystyle \operatorname {tg} ^{2}{\frac {\alpha }{2}}}{\displaystyle 2\,{\operatorname {tg} }\,{\frac {\alpha }{2}}}}.}
ред.
a
sin
x
±
b
cos
x
=
a
2
+
b
2
sin
(
x
±
arcsin
b
a
2
+
b
2
)
,
{\displaystyle a\sin x\pm b\cos x={\sqrt {a^{2}+b^{2}}}\sin \left(x\pm \arcsin {\frac {b}{\sqrt {a^{2}+b^{2}}}}\right),}
a
cos
x
±
b
sin
x
=
a
2
+
b
2
cos
(
x
∓
arccos
a
a
2
+
b
2
)
,
{\displaystyle a\cos x\pm b\sin x={\sqrt {a^{2}+b^{2}}}\cos \left(x\mp \arccos {\frac {a}{\sqrt {a^{2}+b^{2}}}}\right),}
b
+
a
t
g
(
x
)
=
a
2
+
b
2
sin
(
x
+
a
r
c
t
g
(
b
/
a
)
)
cos
x
,
{\displaystyle b+a\mathrm {tg} (x)={\frac {{\sqrt {a^{2}+b^{2}}}\sin(x+\mathrm {arctg} (b/a))}{\cos x}},}
a
+
b
c
t
g
(
x
)
=
a
2
+
b
2
sin
(
x
+
a
r
c
t
g
(
b
/
a
)
)
sin
x
.
{\displaystyle a+b\mathrm {ctg} (x)={\frac {{\sqrt {a^{2}+b^{2}}}\sin(x+\mathrm {arctg} (b/a))}{\sin x}}.}
Перші дві формули можуть бути узагальненими
∑
i
=
1
n
a
i
sin
(
x
+
δ
i
)
=
a
sin
(
x
+
δ
)
,
{\displaystyle \sum _{i=1}^{n}a_{i}\sin(x+\delta _{i})=a\sin(x+\delta ),}
де
a
2
=
∑
i
,
j
=
1
n
a
i
a
j
cos
(
δ
i
−
δ
j
)
,
δ
=
a
r
c
t
g
∑
i
=
1
n
a
i
sin
δ
i
∑
i
=
1
n
a
i
cos
δ
i
.
{\displaystyle a^{2}=\sum _{i,j=1}^{n}a_{i}a_{j}\cos(\delta _{i}-\delta _{j}),\qquad \delta =\mathrm {arctg} \,{\frac {\sum _{i=1}^{n}a_{i}\sin \delta _{i}}{\sum _{i=1}^{n}a_{i}\cos \delta _{i}}}.}
ред.
e
i
x
=
cos
(
x
)
+
i
sin
(
x
)
,
i
2
=
−
1
,
{\displaystyle e^{ix}=\cos(x)+i\sin(x),\,\,i^{2}=-1,}
формула Ейлера ,
ред.
Функція
Обернена функція
sin
θ
=
e
i
θ
−
e
−
i
θ
2
i
{\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}
arcsin
x
=
−
i
ln
(
i
x
+
1
−
x
2
)
{\displaystyle \arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}
cos
θ
=
e
i
θ
+
e
−
i
θ
2
{\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}
arccos
x
=
i
ln
(
x
−
i
1
−
x
2
)
{\displaystyle \arccos x=i\,\ln \left(x-i\,{\sqrt {1-x^{2}}}\right)\,}
tg
θ
=
e
i
θ
−
e
−
i
θ
i
(
e
i
θ
+
e
−
i
θ
)
{\displaystyle \operatorname {tg} \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,}
arctg
x
=
i
2
ln
(
i
+
x
i
−
x
)
{\displaystyle \operatorname {arctg} x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)\,}
csc
θ
=
2
i
e
i
θ
−
e
−
i
θ
{\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\,}
arccsc
x
=
−
i
ln
(
i
x
+
1
−
1
x
2
)
{\displaystyle \operatorname {arccsc} x=-i\ln \left(\displaystyle {\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)\,}
sec
θ
=
2
e
i
θ
+
e
−
i
θ
{\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}\,}
arcsec
x
=
−
i
ln
(
1
x
+
1
−
i
x
2
)
{\displaystyle \operatorname {arcsec} x=-i\ln \left(\displaystyle {\frac {1}{x}}+{\sqrt {1-\displaystyle {\frac {i}{x^{2}}}}}\right)\,}
ctg
θ
=
i
(
e
i
θ
+
e
−
i
θ
)
e
i
θ
−
e
−
i
θ
{\displaystyle \operatorname {ctg} \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,}
arcctg
x
=
i
2
ln
(
x
−
i
x
+
i
)
{\displaystyle \operatorname {arcctg} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)\,}
ред.
sin
2
(
18
∘
)
+
sin
2
(
30
∘
)
=
sin
2
(
36
∘
)
,
{\displaystyle \sin ^{2}(18^{\circ })+\sin ^{2}(30^{\circ })=\sin ^{2}(36^{\circ }),\,}
sin
20
∘
⋅
sin
40
∘
⋅
sin
80
∘
=
3
8
,
{\displaystyle \sin 20^{\circ }\cdot \sin 40^{\circ }\cdot \sin 80^{\circ }={\frac {\sqrt {3}}{8}},}
cos
36
∘
+
cos
108
∘
=
1
2
,
{\displaystyle \cos 36^{\circ }+\cos 108^{\circ }={\frac {1}{2}},}
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}}}
cos
20
∘
⋅
cos
40
∘
⋅
cos
80
∘
=
1
8
,
{\displaystyle \cos 20^{\circ }\cdot \cos 40^{\circ }\cdot \cos 80^{\circ }={\frac {1}{8}},}
∏
k
=
1
n
−
1
sin
(
k
π
n
)
=
n
2
n
−
1
,
{\displaystyle \prod _{k=1}^{n-1}\sin \left({\frac {k\pi }{n}}\right)={\frac {n}{2^{n-1}}},}
∏
k
=
1
n
−
1
cos
(
k
π
n
)
=
sin
(
π
n
/
2
)
2
n
−
1
,
{\displaystyle \prod _{k=1}^{n-1}\cos \left({\frac {k\pi }{n}}\right)={\frac {\sin(\pi n/2)}{2^{n-1}}},}
∏
k
=
1
n
−
1
t
g
(
k
π
n
)
=
n
sin
(
π
n
/
2
)
,
{\displaystyle \prod _{k=1}^{n-1}\mathrm {tg} \,\left({\frac {k\pi }{n}}\right)={\frac {n}{\sin(\pi n/2)}},}
∏
k
=
1
m
t
g
(
k
π
2
m
+
1
)
=
2
m
+
1
,
{\displaystyle \prod _{k=1}^{m}\mathrm {tg} \,\left({\frac {k\pi }{2m+1}}\right)={\sqrt {2m+1}},}
∏
k
=
1
n
sin
(
(
2
k
−
1
)
π
4
n
)
=
∏
k
=
1
n
cos
(
(
2
k
−
1
)
π
4
n
)
=
2
2
n
,
{\displaystyle \prod _{k=1}^{n}\sin \left({\frac {\left(2k-1\right)\pi }{4n}}\right)=\prod _{k=1}^{n}\cos \left({\frac {\left(2k-1\right)\pi }{4n}}\right)={\frac {\sqrt {2}}{2^{n}}},}
π
4
=
4
a
r
c
t
g
1
5
−
a
r
c
t
g
1
239
,
{\displaystyle {\frac {\pi }{4}}=4\mathrm {arctg} \,{\frac {1}{5}}-\mathrm {arctg} \,{\frac {1}{239}},}
π
4
=
5
a
r
c
t
g
1
7
+
2
a
r
c
t
g
3
79
.
{\displaystyle {\frac {\pi }{4}}=5\mathrm {arctg} \,{\frac {1}{7}}+2\mathrm {arctg} \,{\frac {3}{79}}.}
sin
(
π
4
+
α
)
=
cos
(
π
4
−
α
)
,
{\displaystyle \sin \left({\frac {\pi }{4}}+\alpha \right)=\cos \left({\frac {\pi }{4}}-\alpha \right),}
sin
(
π
4
−
α
)
=
cos
(
π
4
+
α
)
,
{\displaystyle \sin \left({\frac {\pi }{4}}-\alpha \right)=\cos \left({\frac {\pi }{4}}+\alpha \right),}
t
g
(
α
+
β
2
)
=
sin
α
+
sin
β
cos
α
+
cos
β
=
−
cos
α
−
cos
β
sin
α
−
sin
β
,
{\displaystyle \mathrm {tg} \left({\frac {\alpha +\beta }{2}}\right)={\frac {\sin \alpha +\sin \beta }{\cos \alpha +\cos \beta }}=-\,{\frac {\cos \alpha -\cos \beta }{\sin \alpha -\sin \beta }},}
1
±
tg
α
=
2
sin
(
π
4
±
α
)
cos
α
,
{\displaystyle 1\pm \operatorname {tg} \alpha ={\frac {{\sqrt {2}}\sin \left(\displaystyle {\frac {\pi }{4}}\pm \alpha \right)}{\cos \alpha }},}
1
±
ctg
α
=
2
sin
(
π
4
±
α
)
sin
α
,
{\displaystyle 1\pm \operatorname {ctg} \alpha ={\frac {{\sqrt {2}}\sin \left(\displaystyle {\frac {\pi }{4}}\pm \alpha \right)}{\sin \alpha }},}
t
g
(
α
)
+
sec
(
α
)
=
tg
(
α
2
+
π
4
)
.
{\displaystyle \mathrm {tg} (\alpha )+\sec(\alpha )=\operatorname {tg} \left({\alpha \over 2}+{\pi \over 4}\right).}
c
t
g
(
α
)
+
tg
(
α
2
)
=
csc
(
α
)
,
{\displaystyle \mathrm {ctg} (\alpha )+\operatorname {tg} \left({\frac {\alpha }{2}}\right)=\csc(\alpha ),}
tg
α
=
sin
2
α
cos
2
α
+
1
,
{\displaystyle \operatorname {tg} \alpha ={\frac {\sin 2\alpha }{\cos 2\alpha +1}},}
tg
5
α
=
tg
α
⋅
tg
(
π
5
+
α
)
⋅
tg
(
π
5
−
α
)
⋅
tg
(
2
π
5
+
α
)
⋅
tg
(
2
π
5
−
α
)
.
{\displaystyle \operatorname {tg} 5\alpha =\operatorname {tg} \alpha \cdot \operatorname {tg} \left({\frac {\pi }{5}}+\alpha \right)\cdot \operatorname {tg} \left({\frac {\pi }{5}}-\alpha \right)\cdot \operatorname {tg} \left({\frac {2\pi }{5}}+\alpha \right)\cdot \operatorname {tg} \left({\frac {2\pi }{5}}-\alpha \right).}
tg
7
α
=
tg
α
⋅
tg
(
π
7
+
α
)
⋅
tg
(
π
7
−
α
)
⋅
tg
(
2
π
7
+
α
)
⋅
tg
(
2
π
7
−
α
)
⋅
tg
(
3
π
7
+
α
)
⋅
tg
(
3
π
7
−
α
)
.
{\displaystyle \operatorname {tg} 7\alpha =\operatorname {tg} \alpha \cdot \operatorname {tg} \left({\frac {\pi }{7}}+\alpha \right)\cdot \operatorname {tg} \left({\frac {\pi }{7}}-\alpha \right)\cdot \operatorname {tg} \left({\frac {2\pi }{7}}+\alpha \right)\cdot \operatorname {tg} \left({\frac {2\pi }{7}}-\alpha \right)\cdot \operatorname {tg} \left({\frac {3\pi }{7}}+\alpha \right)\cdot \operatorname {tg} \left({\frac {3\pi }{7}}-\alpha \right).}
cos
(
α
)
+
2
cos
(
2
α
)
+
3
cos
(
3
α
)
+
⋯
=
∑
k
=
1
n
k
cos
(
k
α
)
=
n
sin
(
α
2
)
sin
(
(
2
n
+
1
)
α
2
)
−
2
sin
2
(
n
α
2
)
2
sin
2
(
α
2
)
{\displaystyle \cos(\alpha )+2\cos(2\alpha )+3\cos(3\alpha )+\cdots =\sum _{k=1}^{n}k\cos(k\alpha )={\frac {\displaystyle n\sin \left({\frac {\alpha }{2}}\right)\sin \left({\frac {(2n+1)\alpha }{2}}\right)-2\sin ^{2}\left({\frac {n\alpha }{2}}\right)}{\displaystyle 2\sin ^{2}\left({\frac {\alpha }{2}}\right)}}}
∑
k
=
1
n
1
2
k
tg
(
α
2
k
)
=
1
2
n
ctg
(
α
2
n
)
−
ctg
α
{\displaystyle \sum _{k=1}^{n}{\frac {1}{2^{k}}}\operatorname {tg} \left({\frac {\alpha }{2^{k}}}\right)={\frac {1}{2^{n}}}\operatorname {ctg} \left({\frac {\alpha }{2^{n}}}\right)-\operatorname {ctg} \alpha }
cos
(
α
)
cos
(
α
2
)
⋅
cos
(
α
4
)
⋯
=
∏
n
=
0
∞
cos
(
α
2
n
)
=
sin
2
α
2
α
.
{\displaystyle \cos(\alpha )\cos \left({\alpha \over 2}\right)\cdot \cos \left({\alpha \over 4}\right)\cdots =\prod \limits _{n=0}^{\infty }\cos \left({\frac {\alpha }{2^{n}}}\right)={\frac {\sin 2\alpha }{2\alpha }}.}
cos
(
α
2
)
⋅
cos
(
α
4
)
⋅
cos
(
α
8
)
⋯
=
∏
n
=
1
∞
cos
(
α
2
n
)
=
sin
α
α
{\displaystyle \cos \left({\alpha \over 2}\right)\cdot \cos \left({\alpha \over 4}\right)\cdot \cos \left({\alpha \over 8}\right)\cdots =\prod _{n=1}^{\infty }\cos \left({\alpha \over 2^{n}}\right)={\sin \alpha \over \alpha }}
∏
k
=
0
n
cos
(
2
k
α
)
=
sin
(
2
n
+
1
α
)
2
n
+
1
sin
α
.
{\displaystyle \prod \limits _{k=0}^{n}\cos \left(2^{k}\alpha \right)={\frac {\sin \left(2^{n+1}\alpha \right)}{2^{n+1}\sin \alpha }}.}
∏
k
=
0
n
cos
(
α
2
k
)
=
sin
2
α
2
n
+
1
sin
(
α
2
n
)
.
{\displaystyle \prod \limits _{k=0}^{n}\cos \left({\frac {\alpha }{2^{k}}}\right)={\frac {\sin 2\alpha }{2^{n+1}\sin \left(\displaystyle {\frac {\alpha }{2^{n}}}\right)}}.}
∏
k
=
1
n
cos
(
α
2
k
)
=
sin
α
2
n
sin
(
α
2
n
)
.
{\displaystyle \prod \limits _{k=1}^{n}\cos \left({\frac {\alpha }{2^{k}}}\right)={\frac {\sin \alpha }{2^{n}\sin \left(\displaystyle {\frac {\alpha }{2^{n}}}\right)}}.}
|
sin
x
|
=
1
2
∏
n
=
0
∞
|
tg
(
2
n
x
)
|
2
n
+
1
.
{\displaystyle |{\sin x}|={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\operatorname {tg} \left(2^{n}x\right)\right|}}.}
Абрамовиц М., Стиган И. Справочник по специальным функциям. — Москва : Наука, 1979. — 832 с.Градштейн И.С., Рыжик И.М. Таблицы интегралов, сумм, рядов и произведений. — Москва : Физматгиз, 1963. — 1100 с. 
![{\displaystyle {\begin{aligned}\mathrm {tg} (\alpha _{1}+\alpha _{2})&={\frac {e_{1}}{e_{0}-e_{2}}}={\frac {x_{1}+x_{2}}{1\ -\ x_{1}x_{2}}}={\frac {\mathrm {tg} \,\alpha _{1}+\mathrm {tg} \,\alpha _{2}}{1\ -\ \mathrm {tg} \,\alpha _{1}\mathrm {tg} \,\alpha _{2}}},\\[8pt]\mathrm {tg} (\alpha _{1}+\alpha _{2}+\alpha _{3})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}}}={\frac {(x_{1}+x_{2}+x_{3})\ -\ (x_{1}x_{2}x_{3})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}}={\frac {\mathrm {tg} \,\alpha _{1}+\mathrm {tg} \,\alpha _{2}+\mathrm {tg} \,\alpha _{3}-\mathrm {tg} \,\alpha _{1}\mathrm {tg} \,\alpha _{2}\mathrm {tg} \,\alpha _{3}}{1-(\mathrm {tg} \,\alpha _{1}\,\mathrm {tg} \,\alpha _{2}+\mathrm {tg} \,\alpha _{1}\,\mathrm {tg} \,\alpha _{3}+\mathrm {tg} \,\alpha _{2}\mathrm {tg} \,\alpha _{3})}},\\[8pt]\mathrm {tg} (\alpha _{1}+\alpha _{2}+\alpha _{3}+\alpha _{4})&={\frac {e_{1}-e_{3}}{e_{0}-e_{2}+e_{4}}}\\[8pt]&={\frac {(x_{1}+x_{2}+x_{3}+x_{4})\ -\ (x_{1}x_{2}x_{3}+x_{1}x_{2}x_{4}+x_{1}x_{3}x_{4}+x_{2}x_{3}x_{4})}{1\ -\ (x_{1}x_{2}+x_{1}x_{3}+x_{1}x_{4}+x_{2}x_{3}+x_{2}x_{4}+x_{3}x_{4})\ +\ (x_{1}x_{2}x_{3}x_{4})}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bd69d1ff8ccfa0792cad858f1f82df4303a170f3)
![{\displaystyle {\begin{aligned}\sec \left(\sum _{i}^{n}\alpha _{i}\right)&={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{e_{0}-e_{2}+e_{4}-\cdots }}={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{\displaystyle \sum _{0\leq 2k\leq n}(-1)^{k}e_{2k}}}\\[8pt]\csc \left(\sum _{i}^{n}\alpha _{i}\right)&={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{e_{1}-e_{3}+e_{5}-\cdots }}={\frac {\displaystyle \prod _{i}^{n}\sec \alpha _{i}}{\displaystyle \sum _{1\leq 2k+1\leq n}(-1)^{k}e_{2k+1}}}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3755d2c2b05e1e090654cd6a0d3f8b154d60a062)
![{\displaystyle {\begin{aligned}\sec(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{1-\mathrm {tg} \,\alpha \,\mathrm {tg} \,\beta -\mathrm {tg} \,\alpha \,\mathrm {tg} \,\gamma -\mathrm {tg} \,\beta \,\mathrm {tg} \,\gamma }},\\[8pt]\csc(\alpha +\beta +\gamma )&={\frac {\sec \alpha \sec \beta \sec \gamma }{\mathrm {tg} \,\alpha +\mathrm {tg} \,\beta +\mathrm {tg} \,\gamma -\mathrm {tg} \,\alpha \,\mathrm {tg} \,\beta \operatorname {tg} \gamma }}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1acbb22bf948da0fd49001113166020cec8b08ae)
![{\displaystyle \sin(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\cos ^{n-2k-1}\alpha \,\sin ^{2k+1}\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/7287eb4afaea326009b665571ac129508e8b20cf)
![{\displaystyle \cos(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\cos ^{n-2k}\alpha \,\sin ^{2k}\alpha }](https://wikimedia.org/api/rest_v1/media/math/render/svg/9faa3f47c0f9ad575ad31758e7c5840475d41ef3)
![{\displaystyle \mathrm {tg} (n\alpha )={\frac {\sin(n\alpha )}{\cos(n\alpha )}}={\dfrac {\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\operatorname {tg} ^{2k+1}\alpha }}{\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\operatorname {tg} ^{2k}\alpha }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/095105e16b5f80a059624241351b7e42bdc2cf6f)
![{\displaystyle \mathrm {ctg} (n\alpha )={\frac {\cos(n\alpha )}{\sin(n\alpha )}}={\dfrac {\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\operatorname {ctg} ^{n-2k}\alpha }}{\displaystyle {\sum \limits _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\operatorname {ctg} ^{n-2k-1}\alpha }}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9250b07edbaaae0cfcd4fb3913f52ea0155cf5cf)
![{\displaystyle [n]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a26847bfc29bbeb4d6ef62ac3fd076378c0fd1db)
![{\displaystyle \left(\cos \alpha +i\sin \alpha \right)^{n}=\sum _{k=0}^{[n/2]}{n \choose 2k}(-1)^{k}(\cos \alpha )^{n-2k}(\sin \alpha )^{2k}+i\left(\sum _{k=0}^{[n/2]}{n \choose 2k+1}(-1)^{k}(\cos \alpha )^{n-2k-1}(\sin \alpha )^{2k+1}\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f4cc47393dea77e38c64e271f33cef901aacaec)
![{\displaystyle \sum _{k=0}^{[n/2]}{n \choose 2k}(-1)^{k}(\cos \alpha )^{n-2k}(\sin \alpha )^{2k}+i\left(\sum _{k=0}^{[n/2]}{n \choose 2k+1}(-1)^{k}(\cos \alpha )^{n-2k-1}(\sin \alpha )^{2k+1}\right)=\cos \left(n\alpha \right)+i\sin \left(n\alpha \right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9efefe3a1be4dbd42ab34f70ad9be92e5d9f128a)
![{\displaystyle \sin(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k+1}}\cos ^{n-2k-1}\alpha \,\sin ^{2k+1}\alpha ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d61f5389af257e4d5471a991d3c99bdb058a384e)
![{\displaystyle \cos(n\alpha )=\sum _{k=0}^{[n/2]}(-1)^{k}{\binom {n}{2k}}\cos ^{n-2k}\alpha \,\sin ^{2k}\alpha .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e66e170e7324a2610635ae5bcc18e4a2aa6169)
![{\displaystyle \sin[\arccos(x)]={\sqrt {1-x^{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/89020e4173e7c25657019ade3629e832bb572ff7)
![{\displaystyle \mathrm {tg} [\arcsin(x)]={\frac {x}{\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/888734314f3d77c7ffe170edfc1690e5b4cc18c7)
![{\displaystyle \sin[\mathrm {arctg} (x)]={\frac {x}{\sqrt {1+x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91f7f9411c2bf1821befa2b1585bbb9a658c4a5a)
![{\displaystyle \mathrm {tg} [\arccos(x)]={\frac {\sqrt {1-x^{2}}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09bb5dce4a2a36dc9f38dd68807c94a0ee8255af)
![{\displaystyle \cos[\mathrm {arctg} (x)]={\frac {1}{\sqrt {1+x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3176d1a4551cfb38ecf06030525c9ab714165780)
![{\displaystyle \mathrm {ctg} [\arcsin(x)]={\frac {\sqrt {1-x^{2}}}{x}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a2a2a4ff52006de67f360a07e37aaec56caecb4)
![{\displaystyle \cos[\arcsin(x)]={\sqrt {1-x^{2}}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6123652c97881e3910234616ea5ee831deae4a73)
![{\displaystyle \mathrm {ctg} [\arccos(x)]={\frac {x}{\sqrt {1-x^{2}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/77ee9fe999f1bc391a66297a49507d912a1b5d35)
![{\displaystyle [-\pi /2,\pi /2]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cd702a5a7041be010f870c0e23750d98ba9919f5)
![{\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/0e0fcfb5b496ea4e16dc633f6858a3982ee07feb)
![{\displaystyle |{\sin x}|={\frac {1}{2}}\prod _{n=0}^{\infty }{\sqrt[{2^{n+1}}]{\left|\operatorname {tg} \left(2^{n}x\right)\right|}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c6e91a4c02e66fcf96c2a0a49642bde42fee1f9)