# Formelsammlung Mathematik: Arkusfunktionen

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### Definition der Arkusfunktionen durch den Logarithmus

${\displaystyle \arcsin z=-\mathrm {i} \,\log \left({\sqrt {1-z^{2}}}+\mathrm {i} z\right)}$

${\displaystyle \arccos z=-\mathrm {i} \,\log \left(z+\mathrm {i} \,{\sqrt {1-z^{2}}}\right)}$

${\displaystyle \arctan z={\frac {\mathrm {i} }{2}}{\Big (}\log(1-\mathrm {i} z)-\log(1+\mathrm {i} z){\Big )}}$ für ${\displaystyle z\neq \pm \mathrm {i} \,}$

${\displaystyle \operatorname {arccot} z=\left\{{\begin{matrix}\arctan \left({\frac {1}{z}}\right)&&z\neq 0\\{\frac {\pi }{2}}&&z=0\end{matrix}}\right.}$

${\displaystyle \operatorname {arcsec} z=\arccos \left({\frac {1}{z}}\right)}$ für ${\displaystyle z\neq 0\,}$

${\displaystyle \operatorname {arccsc} z=\arcsin \left({\frac {1}{z}}\right)}$ für ${\displaystyle z\neq 0\,}$

### Argument iz

${\displaystyle {\begin{matrix}\arcsin(\mathrm {i} z)&=&\;\;\,\mathrm {i} \;\operatorname {arsinh} \,z&\qquad &\operatorname {arsinh} (\mathrm {i} z)&=&\;\;\mathrm {i} \;\arcsin z\\\arctan(\mathrm {i} z)&=&\;\;\,\,\mathrm {i} \;\operatorname {artanh} \,z&\qquad &\operatorname {artanh} (\mathrm {i} z)&=&\;\;\;\mathrm {i} \;\arctan z\\\operatorname {arccot}(\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arcoth} \,z&\qquad &\operatorname {arcoth} (\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arccot} z\\\operatorname {arccsc}(\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arcsch} \,z&\qquad &\operatorname {arcsch} (\mathrm {i} z)&=&-\mathrm {i} \;\operatorname {arccsc} z\\\end{matrix}}}$

### Verkettung einer Winkelfunktion mit einer Arkusfunktion

${\displaystyle f\circ g}$ ${\displaystyle \arcsin \!}$ ${\displaystyle \arccos \!}$ ${\displaystyle \arctan \!}$ ${\displaystyle \operatorname {arccot} \!}$ ${\displaystyle \operatorname {arcsec} \!}$ ${\displaystyle \operatorname {arccsc} \!}$
${\displaystyle \sin \!}$ ${\displaystyle z\!}$ ${\displaystyle {\sqrt {1-z^{2}}}}$ ${\displaystyle {\frac {z}{\sqrt {1+z^{2}}}}}$ ${\displaystyle {\frac {1}{z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}}}$ ${\displaystyle {\sqrt {1-{\frac {1}{z^{2}}}}}}$ ${\displaystyle {\frac {1}{z}}}$
${\displaystyle \cos \!}$ ${\displaystyle {\sqrt {1-z^{2}}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{\sqrt {1+z^{2}}}}}$ ${\displaystyle {\frac {1}{\sqrt {1+{\frac {1}{z^{2}}}}}}}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle {\sqrt {1-{\frac {1}{z^{2}}}}}}$
${\displaystyle \tan \!}$ ${\displaystyle {\frac {z}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {\sqrt {1-z^{2}}}{z}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}$ ${\displaystyle {\frac {1}{z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}}$
${\displaystyle \cot \!}$ ${\displaystyle {\frac {\sqrt {1-z^{2}}}{z}}}$ ${\displaystyle {\frac {z}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}}}$ ${\displaystyle z\,{\sqrt {1-{\frac {1}{z^{2}}}}}}$
${\displaystyle \sec \!}$ ${\displaystyle {\frac {1}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle {\sqrt {1+z^{2}}}}$ ${\displaystyle {\sqrt {1+{\frac {1}{z^{2}}}}}}$ ${\displaystyle z\!}$ ${\displaystyle {\frac {1}{\sqrt {1-{\frac {1}{z^{2}}}}}}}$
${\displaystyle \csc \!}$ ${\displaystyle {\frac {1}{z}}}$ ${\displaystyle {\frac {1}{\sqrt {1-z^{2}}}}}$ ${\displaystyle {\frac {\sqrt {1+z^{2}}}{z}}}$ ${\displaystyle z\,{\sqrt {1+{\frac {1}{z^{2}}}}}}$ ${\displaystyle {\frac {1}{\sqrt {1-{\frac {1}{z^{2}}}}}}}$ ${\displaystyle z\!}$

### Komplementärbeziehungen

${\displaystyle \arcsin(z)+\arccos(z)={\frac {\pi }{2}}}$

${\displaystyle \arctan(z)+\operatorname {arccot}(z)=\left\{{\begin{matrix}{\frac {\pi }{2}}&&\operatorname {Re} (z)>0&&z\in \mathrm {i} \,]-1,0]&&z\in \mathrm {i} \,]1,\infty [\\\\-{\frac {\pi }{2}}&&\operatorname {Re} (z)<0&&z\in \mathrm {i} \,]-\infty ,-1[&&z\in \mathrm {i} \,]0,1[\end{matrix}}\right.}$

${\displaystyle \operatorname {arcsec}(z)+\operatorname {arccsc}(z)={\frac {\pi }{2}}}$ für ${\displaystyle z\neq 0\,}$