info prev up next book cdrom email home

Inverse Cosecant


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

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

\end{displaymath} (1)

for Positive or Negative $x$, and
\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


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

© 1996-9 Eric W. Weisstein