Arccosine of Negative Argument

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: $5.81$