Arccotangent of Negative Argument
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Theorem
Everywhere that the function is defined:
- $\map \arccot {-x} = \pi - \arccot x$
Proof
| \(\ds \map \arccot {-x}\) | \(=\) | \(\ds y\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds -x\) | \(=\) | \(\ds \cot y:\) | \(\ds 0 \le y \le \pi\) | Definition of Arccotangent | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds -\cot y:\) | \(\ds -\pi \le y \le 0\) | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \map \cot {\pi - y}:\) | \(\ds 0 \le y \le \pi\) | Cotangent of Supplementary Angle | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \arccot x\) | \(=\) | \(\ds \pi - y\) | Definition of Arccotangent |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.83$