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