Secant of Supplementary Angle

Theorem

$\map \sec {\pi - \theta} = -\sec \theta$

where $\sec$ denotes secant.

That is, the secant of an angle is the negative of its supplement.

Proof

\(\ds \map \sec {\pi - \theta}\)

\(=\)

\(\ds \frac 1 {\map \cos {\pi - \theta} }\)

Secant is Reciprocal of Cosine

\(\ds \)

\(=\)

\(\ds \frac 1 {-\cos \theta}\)

Cosine of Supplementary Angle

\(\ds \)

\(=\)

\(\ds -\sec \theta\)

Secant is Reciprocal of Cosine

$\blacksquare$

Also see

• Sine of Supplementary Angle
• Cosine of Supplementary Angle
• Tangent of Supplementary Angle
• Cotangent of Supplementary Angle
• Cosecant of Supplementary Angle

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I