Inverse trigonometric functions: Difference between revisions

Revision as of 17:24, 21 May 2024

Revision as of 20:08, 21 May 2024

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\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>

|}

== Applications ==