Editing Inverse trigonometric functions (section)

====Example proof====

:<math>\begin{align}
  \sin(\phi) &= z \\
        \phi &= \arcsin(z)
\end{align}</math>

Using the [[Trigonometric functions#Relationship to exponential function (Euler's formula)|exponential definition of sine]], and letting <math>\xi = e^{i \phi}, </math>

:<math>\begin{align}
z &= \frac{e^{i \phi} - e^{-i \phi}}{2i} \\[10mu]
2iz &= \xi - \frac{1}{\xi} \\[5mu]
0 &= \xi^2 - 2i z \xi - 1 \\[5mu]
\xi &= iz \pm \sqrt{1 - z^2} \\[5mu]
\phi &= -i \ln \left(iz \pm \sqrt{1 - z^2}\right)
\end{align}</math>

(the positive branch is chosen)

:<math>\phi= \arcsin(z) = -i \ln \left(iz + \sqrt{1-z^2} \right)</math>

{| style="text-align:center;"
 |+ [[Domain coloring|Color wheel graphs]] of '''inverse trigonometric functions in the [[complex plane]]'''
 | [[Image:Complex Arcsine.svg|275x275px|Arcsine of z in the complex plane.]]
 | [[Image:Complex Arccosine.svg|275x275px|Arccosine of z in the complex plane.]]
 | [[Image:Complex Arctangent.svg|275x275px|Arctangent of z in the complex plane.]]
 |-
 | <math>\arcsin(z)</math>
 | <math>\arccos(z)</math>
 | <math>\arctan(z)</math>
 |}
{| style="text-align:center;"
 |+
 | [[Image:Complex Arccosecant.svg|275x275px|Arccosecant of z in the complex plane.]]
 | [[Image:Complex Arcsecant.svg|275x275px|Arcsecant of z in the complex plane.]]
 | [[Image:Complex Arccotangent.svg|275x275px|Arccotangent of z in the complex plane.]]
 |-
 | <math>\arccsc(z)</math>
 | <math>\arcsec(z)</math>
 | <math>\arccot(z)</math>
 |}