Arcsecant of Negative Argument

Theorem

Everywhere that the function is defined:

$\map \arcsec {-x} = \pi - \arcsec x$

Proof

\(\ds \map \arcsec {-x}\)

\(=\)

\(\ds y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds -x\)

\(=\)

\(\ds \sec y:\)

\(\ds 0 \le y \le \pi\)

Definition of Arcsecant

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds -\sec y:\)

\(\ds -\pi \le y \le 0\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds \map \sec {\pi - y}:\)

\(\ds 0 \le y \le \pi\)

Cosecant of Supplementary Angle

\(\ds \leadstoandfrom \ \ \)

\(\ds \arcsec x\)

\(=\)

\(\ds \pi - y\)

Definition of Arcsecant

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.84$