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


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
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