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