Arccosine of Negative Argument
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Theorem
Everywhere that the function is defined:
- $\map \arccos {-x} = \pi - \arccos x$
Proof
\(\ds \map \arccos {-x}\) | \(=\) | \(\ds y\) | ||||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds -x\) | \(=\) | \(\ds \cos y:\) | \(\ds 0 \le y \le \pi\) | Definition of Real Arccosine | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds -\cos y:\) | \(\ds -\pi \le y \le 0\) | ||||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \map \cos {\pi - y}:\) | \(\ds 0 \le y \le \pi\) | Cosine of Supplementary Angle | |||||||||
\(\ds \leadstoandfrom \ \ \) | \(\ds \arccos x\) | \(=\) | \(\ds \pi - y\) | Definition of Real Arccosine |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.81$