View source for Inverse trigonometric functions

''inverse circular functions''

Functions inverse to the [[Trigonometric functions|trigonometric functions]]. The six basic trigonometric functions correspond to the six inverse trigonometric functions. These are called arcussine, arcuscosine, arcustangent, arcuscotangent, arcussecant, arcuscosecant, and are denoted, respectively, by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524102.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524106.png" />. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524108.png" /> are defined (in the real domain) for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i0524109.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241011.png" /> for all real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241012.png" />; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241014.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241015.png" />; the last two functions are seldom used. Other notations are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241017.png" />, etc.

Since the trigonometric functions are periodic, their inverses are many-valued. The single-valued branches (principal branches) of these functions are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241019.png" />. Namely, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241020.png" /> is the branch of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241021.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241022.png" />. Similarly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241025.png" /> are defined by the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241028.png" />.

The figures show the graphs of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241031.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241032.png" />; the principal branches are distinguished by a heavy line.

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052410a.gif" />

Figure: i052410a

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052410b.gif" />

Figure: i052410b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052410c.gif" />

Figure: i052410c

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/i052410d.gif" />

Figure: i052410d

The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241033.png" /> are easily expressed in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241034.png" /> for example:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241035.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241036.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241037.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241038.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241039.png" /></td> </tr></table>

The inverse trigonometric functions are related by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241040.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241041.png" /></td> </tr></table>

Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241043.png" /> are not included in the following formulas.

The inverse trigonometric functions are infinitely differentiable and can be expanded in a series in a neighbourhood of any interior point of their domain of definition. The derivatives, integrals and series expansions are:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241044.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241045.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241046.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241047.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241048.png" /></td> </tr></table>

The inverse trigonometric functions of a complex variable are defined as the analytic continuations to the complex plane of the corresponding real functions.

The inverse trigonometric functions can be expressed in terms of the [[Logarithmic function|logarithmic function]]:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241049.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241050.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241051.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241052.png" /></td> </tr></table>



====Comments====
Other notations for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241054.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241055.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i052/i052410/i05241056.png" />, respectively.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.R. Spiegel,   "Complex variables" , ''Schaum's Outline Series'' , McGraw-Hill  (1974)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Abramowitz,   I.A. Stegun,   "Handbook of mathematical functions" , Dover, reprint  (1972)</TD></TR></table>