info prev up next book cdrom email home

Inverse Sine


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

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

\end{displaymath} (1)

for Positive and Negative $x$, and
\sin^{-1}x={\textstyle{1\over 2}}\pi-\sin^{-1}(\sqrt{1-x^2}\,)
\end{displaymath} (2)

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

for Positive or Negative $x$, and
$\displaystyle \sin^{-1}x$ $\textstyle =$ $\displaystyle \cos^{-1}(\sqrt{1-x^2}\,)$ (6)
  $\textstyle =$ $\displaystyle \cot^{-1}\left({\sqrt{1-x^2}\over x}\right)$ (7)
  $\textstyle =$ $\displaystyle \csc^{-1}\left({1\over x}\right)$ (8)
  $\textstyle =$ $\displaystyle \sec^{-1}\left({1\over\sqrt{1-x^2}}\right)$ (9)

for $x\geq 0$.

See also Inverse Cosine, Sine


Abramowitz, M. and Stegun, C. A. (Eds.). ``Inverse Circular Functions.'' §4.4 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 79-83, 1972.

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

© 1996-9 Eric W. Weisstein