Inverse Cosecant

$\begin{figure}\begin{center}\BoxedEPSF{ArcCsc.epsf}\end{center}\end{figure}$

$\begin{figure}\begin{center}\BoxedEPSF{ArcCscReIm.epsf scaled 750}\end{center}\end{figure}$

The function $\csc^{-1}x$ , also denoted arccsc(), where $\csc x$ is the Cosecant and the Superscript denotes an Inverse Function, not the multiplicative inverse. The inverse cosecant satisfies

$\begin{displaymath} \csc^{-1}x=\sec^{-1}\left({x\over\sqrt{x^2-1}}\right) \end{displaymath}$

(1)

for Positive or Negative

, and

$\begin{displaymath} \csc^{-1}x=\pi+\csc^{-1}(-x) \end{displaymath}$

(2)

for $x\geq 0$ . The inverse cosecant is given in terms of other inverse trigonometric functions by

$\displaystyle \csc^{-1}$	$\textstyle =$	$\displaystyle \cos^{-1}\left({\sqrt{x^2-1}\over x}\right)$	(3)
	$\textstyle =$	$\displaystyle \cot^{-1}(\sqrt{x^2-1}\,)$	(4)
	$\textstyle =$	$\displaystyle {\textstyle{1\over 2}}\pi-\sec^{-1}x=-{\textstyle{1\over 2}}\pi-\sec^{-1}(-x)$	(5)
	$\textstyle =$	$\displaystyle \sin^{-1}\left({1\over x}\right)$	(6)

for $x\geq 0$ .

See also Cosecant Inverse Sine, Sine

References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 142-143, 1987.