Inversās trigonometriskās funkcijas jeb ciklometriskās funkcijas ir trigonometrisko funkciju inversās funkcijas . Tām ir sašaurināti definīcijas apgabali , pie tam tā, lai šajā apgabalā katra funkcijas vērtība tiktu iegūta tikai vienu reizi. Pastāv inversā sinusa, kosinusa, tangensa, kotangensa, sekansa un kosekansa funkcijas. Inversās trigonometriskās funkcijas tiek izmantotas, lai aprēķinātu leņķus. Tās plaši tiek izmantotas navigācijā , fizikā , inženierijā u.c.
Visbiežāk inversās trigonometriskās funkcijas pieraksta, parastajai trigonometriskajai funkcijai pieliekot priekšā arc- (latīņu : arcus — ‘loks’) — arcsin x , arccos x utt. Vēl tās var tikt pierakstītas kā sin−1 (x ), cos−1 (x ), tan−1 (x ) utt., taču šajā gadījumā tas var tikt sajaukts ar parasto trigonometrisko funkciju, kas kāpināta −1 pakāpē. Pastāv vēl dažādi to pieraksti.
Nosaukums
Pieraksts
Definīcija
x definīcijas apgabals
Galvenās vērtības diapazons
(radiānos)
Galvenās vērtības diapazons
(grādos)
Arksinuss
y = arcsin x
x = sin y
−1 ≤ x ≤ 1
−π /2 ≤ y ≤ π /2
−90° ≤ y ≤ 90°
Arkkosinuss
y = arccos x
x = cos y
−1 ≤ x ≤ 1
0 ≤ y ≤ π
0° ≤ y ≤ 180°
Arktangenss
y = arctg x
x = tg y
visi reālie skaitļi
−π /2 < y < π /2
−90° < y < 90°
Arkkotangenss
y = arcctg x
x = ctg y
visi reālie skaitļi
0 < y < π
0° < y < 180°
Arksekanss
y = arcsec x
x = sec y
x ≤ −1 vai 1 ≤ x
0 ≤ y < π /2 vai π /2 < y ≤ π
0° ≤ y < 90° vai 90° < y ≤ 180°
Arkkosekanss
y = arccsc x
x = csc y
x ≤ −1 vai 1 ≤ x
−π /2 ≤ y < 0 vai 0 < y ≤ π /2
-90° ≤ y < 0° vai 0° < y ≤ 90°
Atvasinājumi kompleksām z vērtībām.
d
d
z
arcsin
z
=
1
1
−
z
2
;
z
≠
−
1
,
+
1
d
d
z
arccos
z
=
−
1
1
−
z
2
;
z
≠
−
1
,
+
1
d
d
z
arctg
z
=
1
1
+
z
2
;
z
≠
−
i
,
+
i
d
d
z
arcctg
z
=
−
1
1
+
z
2
;
z
≠
−
i
,
+
i
d
d
z
arcsec
z
=
1
z
2
1
−
z
−
2
;
z
≠
−
1
,
0
,
+
1
d
d
z
arccsc
z
=
−
1
z
2
1
−
z
−
2
;
z
≠
−
1
,
0
,
+
1
{\displaystyle {\begin{aligned}{\frac {d}{dz}}\arcsin z&{}={\frac {1}{\sqrt {1-z^{2}}}};\quad z\neq -1,+1\\{\frac {d}{dz}}\arccos z&{}={\frac {-1}{\sqrt {1-z^{2}}}};\quad z\neq -1,+1\\{\frac {d}{dz}}\operatorname {arctg} z&{}={\frac {1}{1+z^{2}}};\quad z\neq -i,+i\\{\frac {d}{dz}}\operatorname {arcctg} z&{}={\frac {-1}{1+z^{2}}};\quad z\neq -i,+i\\{\frac {d}{dz}}\operatorname {arcsec} z&{}={\frac {1}{z^{2}\,{\sqrt {1-z^{-2}}}}};\quad z\neq -1,0,+1\\{\frac {d}{dz}}\operatorname {arccsc} z&{}={\frac {-1}{z^{2}\,{\sqrt {1-z^{-2}}}}};\quad z\neq -1,0,+1\end{aligned}}}
Tikai x reālām vērtībām:
d
d
x
arcsec
x
=
1
|
x
|
x
2
−
1
;
|
x
|
>
1
d
d
x
arccsc
x
=
−
1
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x
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x
2
−
1
;
|
x
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>
1
{\displaystyle {\begin{aligned}{\frac {d}{dx}}\operatorname {arcsec} x&{}={\frac {1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\\{\frac {d}{dx}}\operatorname {arccsc} x&{}={\frac {-1}{|x|\,{\sqrt {x^{2}-1}}}};\qquad |x|>1\end{aligned}}}
arcsin
x
=
∫
0
x
1
1
−
z
2
d
z
,
|
x
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≤
1
arccos
x
=
∫
x
1
1
1
−
z
2
d
z
,
|
x
|
≤
1
arctg
x
=
∫
0
x
1
z
2
+
1
d
z
,
arcctg
x
=
∫
x
∞
1
z
2
+
1
d
z
,
arcsec
x
=
∫
1
x
1
z
z
2
−
1
d
z
,
x
≥
1
arcsec
x
=
π
+
∫
x
−
1
1
z
z
2
−
1
d
z
,
x
≤
−
1
arccsc
x
=
∫
x
∞
1
z
z
2
−
1
d
z
,
x
≥
1
arccsc
x
=
∫
−
∞
x
1
z
z
2
−
1
d
z
,
x
≤
−
1
{\displaystyle {\begin{aligned}\arcsin x&{}=\int _{0}^{x}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\arccos x&{}=\int _{x}^{1}{\frac {1}{\sqrt {1-z^{2}}}}\,dz,\qquad |x|\leq 1\\\operatorname {arctg} x&{}=\int _{0}^{x}{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arcctg} x&{}=\int _{x}^{\infty }{\frac {1}{z^{2}+1}}\,dz,\\\operatorname {arcsec} x&{}=\int _{1}^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arcsec} x&{}=\pi +\int _{x}^{-1}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -1\\\operatorname {arccsc} x&{}=\int _{x}^{\infty }{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\geq 1\\\operatorname {arccsc} x&{}=\int _{-\infty }^{x}{\frac {1}{z{\sqrt {z^{2}-1}}}}\,dz,\qquad x\leq -1\end{aligned}}}
Funkciju izvirzījumi pakāpju rindās :
arcsin
z
=
z
+
(
1
2
)
z
3
3
+
(
1
⋅
3
2
⋅
4
)
z
5
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+
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1
⋅
3
⋅
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2
⋅
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⋅
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)
z
7
7
+
⋯
=
∑
n
=
0
∞
(
2
n
n
)
z
2
n
+
1
4
n
(
2
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+
1
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|
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≤
1
{\displaystyle \arcsin z=z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\left({\frac {1\cdot 3\cdot 5}{2\cdot 4\cdot 6}}\right){\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}
arccos
z
=
π
2
−
arcsin
z
=
π
2
−
(
z
+
(
1
2
)
z
3
3
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⋅
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=
π
2
−
∑
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∞
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z
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{\displaystyle \arccos z={\frac {\pi }{2}}-\arcsin z={\frac {\pi }{2}}-\left(z+\left({\frac {1}{2}}\right){\frac {z^{3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{2n+1}}{4^{n}(2n+1)}};\qquad |z|\leq 1}
arctg
z
=
z
−
z
3
3
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z
5
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⋯
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∑
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0
∞
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−
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2
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+
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≤
1
z
≠
i
,
−
i
{\displaystyle \operatorname {arctg} z=z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}
arcctg
z
=
π
2
−
arctg
z
=
π
2
−
(
z
−
z
3
3
+
z
5
5
−
z
7
7
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⋯
)
=
π
2
−
∑
n
=
0
∞
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−
1
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z
2
n
+
1
2
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+
1
;
|
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≤
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z
≠
i
,
−
i
{\displaystyle \operatorname {arcctg} z={\frac {\pi }{2}}-\operatorname {arctg} z\ ={\frac {\pi }{2}}-\left(z-{\frac {z^{3}}{3}}+{\frac {z^{5}}{5}}-{\frac {z^{7}}{7}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {(-1)^{n}z^{2n+1}}{2n+1}};\qquad |z|\leq 1\qquad z\neq i,-i}
arcsec
z
=
arccos
(
1
/
z
)
=
π
2
−
(
z
−
1
+
(
1
2
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z
−
3
3
+
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1
⋅
3
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⋅
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z
−
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⋯
)
=
π
2
−
∑
n
=
0
∞
(
2
n
n
)
z
−
(
2
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+
1
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4
n
(
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+
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≥
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{\displaystyle \operatorname {arcsec} z=\arccos {(1/z)}={\frac {\pi }{2}}-\left(z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ \right)={\frac {\pi }{2}}-\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}
arccsc
z
=
arcsin
(
1
/
z
)
=
z
−
1
+
(
1
2
)
z
−
3
3
+
(
1
⋅
3
2
⋅
4
)
z
−
5
5
+
⋯
=
∑
n
=
0
∞
(
2
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)
z
−
(
2
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+
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)
4
n
(
2
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+
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z
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≥
1
{\displaystyle \operatorname {arccsc} z=\arcsin {(1/z)}=z^{-1}+\left({\frac {1}{2}}\right){\frac {z^{-3}}{3}}+\left({\frac {1\cdot 3}{2\cdot 4}}\right){\frac {z^{-5}}{5}}+\cdots \ =\sum _{n=0}^{\infty }{\frac {{\binom {2n}{n}}z^{-(2n+1)}}{4^{n}(2n+1)}};\qquad |z|\geq 1}
Reālām un kompleksām x vērtībām:
∫
arcsin
x
d
x
=
x
arcsin
x
+
1
−
x
2
+
C
∫
arccos
x
d
x
=
x
arccos
x
−
1
−
x
2
+
C
∫
arctg
x
d
x
=
x
arctg
x
−
1
2
ln
(
1
+
x
2
)
+
C
∫
arcctg
x
d
x
=
x
arcctg
x
+
1
2
ln
(
1
+
x
2
)
+
C
∫
arcsec
x
d
x
=
x
arcsec
x
−
ln
[
x
(
1
+
x
2
−
1
x
2
)
]
+
C
∫
arccsc
x
d
x
=
x
arccsc
x
+
ln
[
x
(
1
+
x
2
−
1
x
2
)
]
+
C
{\displaystyle {\begin{aligned}\int \arcsin x\,dx&{}=x\,\arcsin x+{\sqrt {1-x^{2}}}+C\\\int \arccos x\,dx&{}=x\,\arccos x-{\sqrt {1-x^{2}}}+C\\\int \operatorname {arctg} x\,dx&{}=x\,\operatorname {arctg} x-{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arcctg} x\,dx&{}=x\,\operatorname {arcctg} x+{\frac {1}{2}}\ln \left(1+x^{2}\right)+C\\\int \operatorname {arcsec} x\,dx&{}=x\,\operatorname {arcsec} x-\ln \left[x\left(1+{\sqrt {{x^{2}-1} \over x^{2}}}\right)\right]+C\\\int \operatorname {arccsc} x\,dx&{}=x\,\operatorname {arccsc} x+\ln \left[x\left(1+{\sqrt {{x^{2}-1} \over x^{2}}}\right)\right]+C\end{aligned}}}
Reālām un kompleksām x ≥ 1 vērtībām:
∫
arcsec
x
d
x
=
x
arcsec
x
−
ln
(
x
+
x
2
−
1
)
+
C
∫
arccsc
x
d
x
=
x
arccsc
x
+
ln
(
x
+
x
2
−
1
)
+
C
{\displaystyle {\begin{aligned}\int \operatorname {arcsec} x\,dx&{}=x\,\operatorname {arcsec} x-\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\\\int \operatorname {arccsc} x\,dx&{}=x\,\operatorname {arccsc} x+\ln \left(x+{\sqrt {x^{2}-1}}\right)+C\end{aligned}}}