Inverse Tangent is Odd Function
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Theorem
Everywhere that the function is defined:
- $\map \arctan {-x} = -\arctan x$
Proof
\(\ds \map \arctan {-x}\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds -x\) | \(=\) | \(\ds \tan y:\) | \(\ds -\frac \pi 2 \le y \le \frac \pi 2\) | Definition of Real Arctangent | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds -\tan y:\) | \(\ds -\frac \pi 2 \le y \le \frac \pi 2\) | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \map \tan {-y}:\) | \(\ds -\frac \pi 2 \le y \le \frac \pi 2\) | Tangent Function is Odd | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \arctan x\) | \(=\) | \(\ds -y\) | Definition of Real Arctangent |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.82$