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Inverse Cosine


\begin{figure}\begin{center}\BoxedEPSF{ArcCosReIm.epsf scaled 770}\end{center}\end{figure}

The function $\cos^{-1} x$, also denoted arccos($x$), where $\cos x$ is the Cosine and the superscript $-1$ denotes an Inverse Function, not the multiplicative inverse. The Maclaurin Series for the inverse cosine range $-1<x<1$ is

\cos^{-1} x = {\textstyle{1\over 2}}\pi-x-{\textstyle{1\over...
...xtstyle{5\over 112}}x^7 -{\textstyle{35\over 1152}}x^9-\ldots.
\end{displaymath} (1)

The inverse cosine satisfies
\end{displaymath} (2)

for Positive and Negative $x$, and
\cos^{-1}x={\textstyle{1\over 2}}\pi-\cos^{-1}(\sqrt{1-x^2}\,)
\end{displaymath} (3)

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

for Positive or Negative $x$, and
$\displaystyle \cos^{-1}x$ $\textstyle =$ $\displaystyle \csc^{-1}\left({1\over\sqrt{1-x^2}}\right)$ (7)
  $\textstyle =$ $\displaystyle \sec^{-1}\left({1\over x}\right)$ (8)
  $\textstyle =$ $\displaystyle \sin^{-1}(\sqrt{1-x^2}\,)$ (9)
  $\textstyle =$ $\displaystyle \tan^{-1}\left({\sqrt{1-x^2}\over x}\right)$ (10)

for $x\geq 0$.

See also Cosine, Inverse Secant


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.

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© 1996-9 Eric W. Weisstein