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 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.
Inverso trigonometrisko funkciju uzskaitījums Labot
Atvasinājumi Labot
Atvasinājumi kompleksām z vērtībām.
d
d
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arcsin
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1
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arccos
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arctg
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{\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:
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{\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}}}
Noteiktie, neīstie integrāļi Labot
arcsin
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arccos
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arctg
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{\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 Labot
Funkciju izvirzījumi pakāpju rindās :
arcsin
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{\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
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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
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{\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
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{\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
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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
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{\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}
Nenoteiktie integrāļi Labot
Reālām un kompleksām x vērtībām:
∫
arcsin
x
d
x
=
x
arcsin
x
+
1
−
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2
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∫
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{\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
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−
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{\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 .
Ārējās saites Labot 
![{\displaystyle \left[-{\frac {\pi }{2}};{\frac {\pi }{2}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8953e7a70dae484772b5a2b5cab77216226235d)
![{\displaystyle \left[0;\pi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb479d863a2d0b151a6ea8334e4ab6189b7a759)
![{\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aeebcec6085490340dd4f8dc779b9d6166b49c5a)
