Editing Inverse trigonometric functions (section)

== Extension to the complex plane ==

[[File:Riemann surface for Arg of ArcTan of x.svg|thumb|A [[Riemann surface]] for the argument of the relation {{math|1=tan ''z'' = ''x''}}. The orange sheet in the middle is the principal sheet representing {{math|arctan ''x''}}. The blue sheet above and green sheet below are displaced by {{math|2''&pi;''}} and {{math|−2''&pi;''}} respectively.]]

Since the inverse trigonometric functions are [[analytic function]]s, they can be extended from the real line to the complex plane. This results in functions with multiple sheets and [[branch point]]s. One possible way of defining the extension is:
:<math>\arctan(z) = \int_0^z \frac{dx}{1 + x^2} \quad z \neq -i, +i </math>
where the part of the imaginary axis which does not lie strictly between the branch points (−i and +i) is the [[branch cut]] between the principal sheet and other sheets. The path of the integral must not cross a branch cut. For ''z'' not on a branch cut, a straight line path from 0 to ''z'' is such a path. For ''z'' on a branch cut, the path must approach from {{nowrap|Re[x] > 0}} for the upper branch cut and from {{nowrap|Re[x] < 0}} for the lower branch cut.

The arcsine function may then be defined as:

:<math>\arcsin(z) = \arctan\left(\frac{z}{\sqrt{1 - z^2}}\right) \quad z \neq -1, +1 </math>
where (the square-root function has its cut along the negative real axis and) the part of the real axis which does not lie strictly between &minus;1 and +1 is the branch cut between the principal sheet of arcsin and other sheets;
:<math>\arccos(z) = \frac{\pi}{2} - \arcsin(z) \quad z \neq -1, +1 </math>
which has the same cut as arcsin;
:<math>\arccot(z) = \frac{\pi}{2} - \arctan(z) \quad z \neq -i, i </math>
which has the same cut as arctan;
:<math>\arcsec(z) = \arccos\left(\frac{1}{z}\right) \quad z \neq -1, 0, +1 </math>
where the part of the real axis between &minus;1 and +1 inclusive is the cut between the principal sheet of arcsec and other sheets;
:<math>\arccsc(z) = \arcsin\left(\frac{1}{z}\right) \quad z \neq -1, 0, +1 </math>
which has the same cut as arcsec.

===Logarithmic forms===
These functions may also be expressed using [[complex logarithm]]s. This extends their [[domain of a function|domains]] to the [[complex plane]] in a natural fashion. The following identities for principal values of the functions hold everywhere that they are defined, even on their branch cuts.

:<math>\begin{align}
\arcsin(z) &{}= -i \ln \left( \sqrt{1-z^2} + iz \right) = i \ln \left( \sqrt{1-z^2} - iz \right) &{}= \arccsc\left(\frac{1}{z}\right) \\[10pt]
\arccos(z) &{}= -i \ln \left( i \sqrt {1-z^2} + z \right) = \frac{\pi}{2} - \arcsin(z) &{}= \arcsec\left(\frac{1}{z}\right) \\[10pt]
\arctan(z) &{}= -\frac{i}{2}\ln \left(\frac{i - z}{i + z}\right) = -\frac{i}{2}\ln \left(\frac{1 + iz}{1 - iz}\right) &{}= \arccot\left(\frac{1}{z}\right) \\[10pt]
\arccot(z) &{}= -\frac{i}{2}\ln\left( \frac{z + i}{z - i} \right) = -\frac{i}{2}\ln\left( \frac{iz - 1}{iz + 1} \right) &{}= \arctan\left(\frac{1}{z}\right) \\[10pt]
\arcsec(z) &{}= -i \ln \left( i \sqrt{1 - \frac{1}{z^2}} + \frac{1}{z} \right) = \frac{\pi}{2} - \arccsc(z) &{}= \arccos\left(\frac{1}{z}\right) \\[10pt]
\arccsc(z) &{}= -i \ln \left( \sqrt{1 - \frac{1}{z^2}} + \frac{i}{z} \right) = i \ln \left( \sqrt{1 - \frac{1}{z^2}} - \frac{i}{z} \right) &{}= \arcsin\left(\frac{1}{z}\right)
\end{align}</math>

====Generalization====

Because all of the inverse trigonometric functions output an angle of a right triangle, they can be generalized by using [[Euler's formula]] to form a right triangle in the complex plane. Algebraically, this gives us:

:<math>ce^{i\theta} = c\cos(\theta) + ic\sin(\theta)</math>
or
:<math>ce^{i\theta} = a + ib</math>

where <math>a</math> is the adjacent side, <math>b</math> is the opposite side, and <math>c</math> is the hypotenuse. From here, we can solve for <math>\theta</math>.

:<math>\begin{align}
e^{\ln(c) + i\theta} & = a + ib \\
\ln c + i\theta & = \ln(a + ib) \\
\theta & = \operatorname{Im}\left( \ln(a + ib) \right)
\end{align}</math>
or
:<math>\theta = -i\ln\left(\frac{a + ib}{c}\right)</math>

Simply taking the imaginary part works for any real-valued <math>a</math> and <math>b</math>, but if <math>a</math> or <math>b</math> is complex-valued, we have to use the final equation so that the real part of the result isn't excluded. Since the length of the hypotenuse doesn't change the angle, ignoring the real part of <math>\ln(a+bi)</math> also removes <math>c</math> from the equation. In the final equation, we see that the angle of the triangle in the complex plane can be found by inputting the lengths of each side. By setting one of the three sides equal to 1 and one of the remaining sides equal to our input <math>z</math>, we obtain a formula for one of the inverse trig functions, for a total of six equations. Because the inverse trig functions require only one input, we must put the final side of the triangle in terms of the other two using the [[Pythagorean Theorem]] relation
:<math>a^2 + b^2 = c^2</math>

The table below shows the values of a, b, and c for each of the inverse trig functions and the equivalent expressions for <math>\theta</math> that result from plugging the values into the equations <math>\theta = -i\ln\left(\tfrac{a + ib}{c}\right)</math> above and simplifying.
:<math>\begin{align}
 & a & & b & & c && -i\ln\left(\frac{a+ib}{c}\right) && \theta && \theta_{a,b\in\R}\\
\arcsin(z)\ \ & \sqrt{1 - z^2} & & z & & 1 & & -i\ln\left( \frac{\sqrt{1 - z^2} + iz}{1} \right) && = -i\ln\left( \sqrt{1 - z^2} + iz \right) && \operatorname{Im}\left(\ln\left( \sqrt{1 - z^2} + iz \right)\right) \\
\arccos(z)\ \ & z & & \sqrt{1 - z^2} & & 1 & & -i\ln\left( \frac{z + i\sqrt{1 - z^2}}{1} \right) && = -i\ln\left( z + \sqrt{z^2 - 1} \right) && \operatorname{Im}\left(\ln\left( z + \sqrt{z^2 - 1} \right)\right) \\
\arctan(z)\ \ & 1 & & z & & \sqrt{1 + z^2} & & -i\ln\left( \frac{1 + iz}{\sqrt{1 + z^2}} \right) && = -\frac{i}{2}\ln\left( \frac{i-z}{i+z} \right) && \operatorname{Im}\left(\ln\left( 1 + iz \right)\right) \\
\arccot(z)\ \ & z & & 1 & & \sqrt{z^2 + 1} & & -i\ln\left( \frac{z + i}{\sqrt{z^2 + 1}} \right) && = -\frac{i}{2}\ln\left( \frac{z + i}{z-i} \right) && \operatorname{Im}\left(\ln\left( z + i \right)\right) \\
\arcsec(z)\ \ & 1 & & \sqrt{z^2 - 1} & & z & & -i\ln\left( \frac{1 + i\sqrt{z^2 - 1}}{z} \right) && = -i\ln\left( \frac{1}{z} + \sqrt{\frac{1}{z^2}-1} \right) && \operatorname{Im}\left(\ln\left( \frac{1}{z} + \sqrt{\frac{1}{z^2}-1} \right)\right) \\
\arccsc(z)\ \ & \sqrt{z^2 - 1} & & 1 & & z & & -i\ln\left( \frac{\sqrt{z^2 - 1} + i}{z} \right) && = -i\ln\left( \sqrt{1-\frac{1}{z^2}} + \frac{i}{z} \right) && \operatorname{Im}\left(\ln\left( \sqrt{1 - \frac{1}{z^2}} + \frac{i}{z} \right)\right) \\
\end{align}</math>

The particular form of the simplified expression can cause the output to differ from the [[#Principal_values|usual principal branch]] of each of the inverse trig functions. The formulations given will output the usual principal branch when using the <math> \operatorname{Im}\left( \ln z \right) \in (-\pi,\pi] </math> and <math> \operatorname{Re}\left(\sqrt{z}\right) \ge 0 </math> principal branch for every function except arccotangent in the <math>\theta</math> column. Arccotangent in the <math>\theta</math> column will output on its usual principal branch by using the <math> \operatorname{Im}\left( \ln z \right) \in [0,2\pi) </math> and <math> \operatorname{Im}\left(\sqrt{z}\right) \ge 0 </math> convention.

In this sense, all of the inverse trig functions can be thought of as specific cases of the complex-valued log function. Since these definition work for any complex-valued <math>z</math>, the definitions allow for [[hyperbolic angle]]s as outputs and can be used to further define the [[inverse hyperbolic functions]]. It's possible to algebraically prove these relations by starting with the exponential forms of the trigonometric functions and solving for the inverse function.

====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>
 |}