21. Exponential definitions

Function	Inverse function[46]
${\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}\,}$	${\displaystyle \arcsin x=-i\ln \left(ix+{\sqrt {1-x^{2}}}\right)\,}$
${\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}\,}$	${\displaystyle \arccos x=i\,\ln \left(x-i\,{\sqrt {1-x^{2}}}\right)\,}$
${\displaystyle \tan \theta ={\frac {e^{i\theta }-e^{-i\theta }}{i(e^{i\theta }+e^{-i\theta })}}\,}$	${\displaystyle \arctan x={\frac {i}{2}}\ln \left({\frac {i+x}{i-x}}\right)\,}$
${\displaystyle \csc \theta ={\frac {2i}{e^{i\theta }-e^{-i\theta }}}\,}$	${\displaystyle \operatorname {arccsc} x=-i\ln \left({\frac {i}{x}}+{\sqrt {1-{\frac {1}{x^{2}}}}}\right)\,}$
${\displaystyle \sec \theta ={\frac {2}{e^{i\theta }+e^{-i\theta }}}\,}$	${\displaystyle \operatorname {arcsec} x=-i\ln \left({\frac {1}{x}}+{\sqrt {1-{\frac {i}{x^{2}}}}}\right)\,}$
${\displaystyle \cot \theta ={\frac {i(e^{i\theta }+e^{-i\theta })}{e^{i\theta }-e^{-i\theta }}}\,}$	${\displaystyle \operatorname {arccot} x={\frac {i}{2}}\ln \left({\frac {x-i}{x+i}}\right)\,}$
${\displaystyle \operatorname {cis} \,\theta =e^{i\theta }\,}$	${\displaystyle \operatorname {arccis} \,x={\frac {\ln x}{i}}=-i\ln x=\operatorname {arg} \,x\,}$