Inverse Cosine

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

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

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

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

(1)

The inverse cosine satisfies

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

(2)

for Positive and Negative

, and

$\begin{displaymath} \cos^{-1}x={\textstyle{1\over 2}}\pi-\cos^{-1}(\sqrt{1-x^2}\,) \end{displaymath}$

(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

, 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

References

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.