Secant of Conjugate Angle

Theorem

$\map \sec {2 \pi - \theta} = \sec \theta$

where $\sec$ denotes secant.

That is, the secant of an angle equals its conjugate.

Proof

\(\ds \map \sec {2 \pi - \theta}\)

\(=\)

\(\ds \frac 1 {\map \cos {2 \pi - \theta} }\)

Secant is Reciprocal of Cosine

\(\ds \)

\(=\)

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

Cosine of Conjugate Angle

\(\ds \)

\(=\)

\(\ds \sec \theta\)

Secant is Reciprocal of Cosine

$\blacksquare$

Examples

Secant of $360 \degrees - 3 x$

$\map \sec {360 \degrees - 3 x} = \sec 3 x$

Also see

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