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Inverse Secant


\begin{figure}\begin{center}\BoxedEPSF{ArcSecReIm.epsf scaled 700}\end{center}\end{figure}

The function $\sec^{-1}x$, where $\sec x$ is the Secant and the superscript $-1$ denotes the Inverse Function, not the multiplicative inverse. The inverse secant satisfies

\end{displaymath} (1)

for Positive or Negative $x$, and
\end{displaymath} (2)

for $x\geq 0$. The inverse secant is given in terms of other inverse trigonometric functions by
$\displaystyle \sec^{-1}x$ $\textstyle =$ $\displaystyle \cos^{-1}\left({1\over x}\right)$ (3)
  $\textstyle =$ $\displaystyle \cot^{-1}\left({1\over\sqrt{x^2-1}}\right)$ (4)
  $\textstyle =$ $\displaystyle {\textstyle{1\over 2}}\pi-\csc^{-1}x=-{\textstyle{1\over 2}}\pi-\csc^{-1}(-x)$ (5)
  $\textstyle =$ $\displaystyle \sin^{-1}\left({\sqrt{x^2-1}\over x}\right)$ (6)
  $\textstyle =$ $\displaystyle \tan^{-1}(\sqrt{x^2-1}\,)$ (7)

for $x\geq 0$.

See also Inverse Cosecant, Secant


Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 141-143, 1987.

© 1996-9 Eric W. Weisstein