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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 ${\displaystyle z\neq 0\,}$