# Inverse Sine is Odd Function

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## Theorem

Everywhere that the function is defined:

- $\map \arcsin {-x} = -\arcsin x$

## Proof

\(\ds \map \arcsin {-x}\) | \(=\) | \(\ds y\) | ||||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds -x\) | \(=\) | \(\ds \sin y:\) | \(\ds -\frac \pi 2 \le y \le \frac \pi 2\) | Definition of Arcsine | |||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds -\sin y:\) | \(\ds -\frac \pi 2 \le y \le \frac \pi 2\) | ||||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \map \sin {-y}:\) | \(\ds -\frac \pi 2 \le y \le \frac \pi 2\) | Sine Function is Odd | |||||||||

\(\ds \leadstoandfrom \ \ \) | \(\ds \arcsin x\) | \(=\) | \(\ds -y\) | Definition of Arcsine |

$\blacksquare$

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.80$