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°
arcsin a
Intervāla
[
−
π
2
;
π
2
]
{\displaystyle \left[-{\frac {\pi }{2}};{\frac {\pi }{2}}\right]}
leņķis, kura sinusa funkcijas vērtība ir skaitlis a (|a |≤1). [1]
Piemērs :
a
r
c
s
i
n
1
2
=
π
6
{\displaystyle arcsin{\frac {1}{2}}={\frac {\pi }{6}}}
, jo
s
i
n
π
6
=
1
2
{\displaystyle sin{\frac {\pi }{6}}={\frac {1}{2}}}
un
π
6
{\displaystyle {\frac {\pi }{6}}}
∈
{\displaystyle \in }
[
−
π
2
;
π
2
]
{\displaystyle \left[-{\frac {\pi }{2}};{\frac {\pi }{2}}\right]}
Biežāk sastopamo arksinusa vērtību tabula[2]
Funkcija
Arguments a
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
−
1
2
{\displaystyle -{\frac {1}{2}}}
1
2
{\displaystyle {\frac {1}{2}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
arcsin a
−
π
3
{\displaystyle -{\tfrac {\pi }{3}}}
−
π
4
{\displaystyle -{\tfrac {\pi }{4}}}
−
π
6
{\displaystyle -{\tfrac {\pi }{6}}}
π
6
{\displaystyle {\tfrac {\pi }{6}}}
π
4
{\displaystyle {\tfrac {\pi }{4}}}
π
3
{\displaystyle {\tfrac {\pi }{3}}}
arccos a
Intervāla
[
0
;
π
]
{\displaystyle \left[0;\pi \right]}
leņķis, kura kosinusa funkcijas vērtība ir skaitlis a (|a |≤1). [1]
Piemērs :
a
r
c
c
o
s
2
2
=
π
4
{\displaystyle arccos{\frac {\sqrt {2}}{2}}={\frac {\pi }{4}}}
, jo
c
o
s
π
4
=
2
2
{\displaystyle cos{\frac {\pi }{4}}={\frac {\sqrt {2}}{2}}}
un
π
4
{\displaystyle {\frac {\pi }{4}}}
∈
{\displaystyle \in }
[
0
;
π
]
{\displaystyle \left[0;\pi \right]}
Biežāk sastopamo arkkosinusa vērtību tabula [2]
Funkcija
Arguments a
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
−
3
2
{\displaystyle -{\frac {\sqrt {3}}{2}}}
−
1
2
{\displaystyle -{\frac {1}{2}}}
1
2
{\displaystyle {\frac {1}{2}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
arccos a
5
π
6
{\displaystyle {\frac {5\pi }{6}}}
3
π
4
{\displaystyle {\frac {3\pi }{4}}}
2
π
3
{\displaystyle {\frac {2\pi }{3}}}
π
3
{\displaystyle {\frac {\pi }{3}}}
π
4
{\displaystyle {\frac {\pi }{4}}}
π
6
{\displaystyle {\frac {\pi }{6}}}
arctg a
Intervāla
[
−
π
2
;
π
2
]
{\displaystyle \left[-{\frac {\pi }{2}};{\frac {\pi }{2}}\right]}
leņķis, kura tangensa funkcijas vērtība ir skaitlis a .[1]
Piemērs :
a
r
c
t
g
3
=
π
3
{\displaystyle arctg{\sqrt {3}}={\frac {\pi }{3}}}
, jo
t
g
π
3
=
3
{\displaystyle tg{\frac {\pi }{3}}={\sqrt {3}}}
un
π
3
{\displaystyle {\frac {\pi }{3}}}
∈
{\displaystyle \in }
[
−
π
2
;
π
2
]
{\displaystyle \left[-{\frac {\pi }{2}};{\frac {\pi }{2}}\right]}
Biežāk sastopamo arktangensa vērtību tabula[2]
Funkcija
Arguments a
−
3
{\displaystyle -{\sqrt {3}}}
−
1
{\displaystyle -1}
−
3
3
{\displaystyle -{\frac {\sqrt {3}}{3}}}
0
{\displaystyle 0}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
1
{\displaystyle 1}
3
{\displaystyle {\sqrt {3}}}
arctg a
−
π
3
{\displaystyle -{\tfrac {\pi }{3}}}
−
π
4
{\displaystyle -{\tfrac {\pi }{4}}}
−
π
6
{\displaystyle -{\tfrac {\pi }{6}}}
0
{\displaystyle 0}
π
6
{\displaystyle {\tfrac {\pi }{6}}}
π
4
{\displaystyle {\tfrac {\pi }{4}}}
π
3
{\displaystyle {\tfrac {\pi }{3}}}
arcctg a
Intervāla
[
0
;
π
]
{\displaystyle \left[0;\pi \right]}
leņķis, kura kotangensa funkcijas vērtība ir skaitlis a .[1]
Piemērs :
a
r
c
c
t
g
1
=
π
4
{\displaystyle arcctg1={\frac {\pi }{4}}}
, jo
c
t
g
π
4
=
1
{\displaystyle ctg{\frac {\pi }{4}}=1}
un
π
4
{\displaystyle {\frac {\pi }{4}}}
∈
{\displaystyle \in }
[
0
;
π
]
{\displaystyle \left[0;\pi \right]}
Biežāk sastopamo arkkotangensa vērtību tabula[2]
Funkcija
Arguments a
−
3
{\displaystyle -{\sqrt {3}}}
−
1
{\displaystyle -1}
−
3
3
{\displaystyle -{\frac {\sqrt {3}}{3}}}
0
{\displaystyle 0}
3
3
{\displaystyle {\frac {\sqrt {3}}{3}}}
1
{\displaystyle 1}
3
{\displaystyle {\sqrt {3}}}
arcctg a
5
π
6
{\displaystyle {\frac {5\pi }{6}}}
3
π
4
{\displaystyle {\frac {3\pi }{4}}}
2
π
3
{\displaystyle {\frac {2\pi }{3}}}
π
2
{\displaystyle {\frac {\pi }{2}}}
π
3
{\displaystyle {\frac {\pi }{3}}}
π
4
{\displaystyle {\frac {\pi }{4}}}
π
6
{\displaystyle {\frac {\pi }{6}}}
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
|
x
|
x
2
−
1
;
|
x
|
>
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
|
≤
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
5
+
(
1
⋅
3
⋅
5
2
⋅
4
⋅
6
)
z
7
7
+
⋯
=
∑
n
=
0
∞
(
2
n
n
)
z
2
n
+
1
4
n
(
2
n
+
1
)
;
|
z
|
≤
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
+
(
1
⋅
3
2
⋅
4
)
z
5
5
+
⋯
)
=
π
2
−
∑
n
=
0
∞
(
2
n
n
)
z
2
n
+
1
4
n
(
2
n
+
1
)
;
|
z
|
≤
1
{\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
+
z
5
5
−
z
7
7
+
⋯
=
∑
n
=
0
∞
(
−
1
)
n
z
2
n
+
1
2
n
+
1
;
|
z
|
≤
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
+
⋯
)
=
π
2
−
∑
n
=
0
∞
(
−
1
)
n
z
2
n
+
1
2
n
+
1
;
|
z
|
≤
1
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
)
z
−
3
3
+
(
1
⋅
3
2
⋅
4
)
z
−
5
5
+
⋯
)
=
π
2
−
∑
n
=
0
∞
(
2
n
n
)
z
−
(
2
n
+
1
)
4
n
(
2
n
+
1
)
;
|
z
|
≥
1
{\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
n
n
)
z
−
(
2
n
+
1
)
4
n
(
2
n
+
1
)
;
|
z
|
≥
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}}}
↑ 1,0 1,1 1,2 1,3 Kārlis Šteiners, Biruta Siliņa. Rokasgrāmata matemātikā . Rīga : Zvaigzne ABC, 2006. ISBN 978-9984-40-584-1 .
↑ 2,0 2,1 2,2 2,3 D. Kriķis, P.Zariņš, V.Ziobrovskis. Diferencēti uzdevumi matemātikā . Rīga : Zvaigzne ABC, 1996. ISBN 5-405-01338-2 .