Inverse Sine

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

$$\sin^{-1}x=-\sin^{-1}(-x)$$ (1)

for Positive and Negative $x$, and

$$\sin^{-1}x={\textstyle{1\over 2}}\pi-\sin^{-1}(\sqrt{1-x^2}\,)$$ (2)

for $x\geq 0$. The inverse sine is given in terms of other inverse trigonometric functions by

$$\sin^{-1}x = \cos^{-1}(-x)-{\textstyle{1\over 2}}\pi={\textstyle{1\over 2}}\pi-\cos^{-1}x$$ (3)

$$= {\textstyle{1\over 2}}\pi-\cot^{-1}\left({x\over\sqrt{1-x^2}}\right)$$ (4)

$$= \tan^{-1}\left({x\over\sqrt{1-x^2}}\right)$$ (5)

for Positive or Negative $x$, and

$$\sin^{-1}x = \cos^{-1}(\sqrt{1-x^2}\,)$$ (6)

$$= \cot^{-1}\left({\sqrt{1-x^2}\over x}\right)$$ (7)

$$= \csc^{-1}\left({1\over x}\right)$$ (8)

$$= \sec^{-1}\left({1\over\sqrt{1-x^2}}\right)$$ (9)

for $x\geq 0$.

See also Inverse Cosine, Sine

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.