# Inverse Sine is Odd Function

## 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

$\blacksquare$