Particular Values of Secant Function

Theorem

The following values of the secant function can be expressed as exact algebraic numbers.

This list is non-exhaustive.

Secant of Zero

$\sec 0 = 0$

Secant of $15 \degrees$

$\sec 15 \degrees = \sec \dfrac \pi {12} = \sqrt 6 - \sqrt 2$

Secant of $30 \degrees$

$\sec 30 \degrees = \sec \dfrac \pi 6 = \dfrac {2 \sqrt 3} 3$

Secant of $45 \degrees$

$\sec 45 \degrees = \sec \dfrac \pi 4 = \sqrt 2$

Secant of $60 \degrees$

$\sec 60 \degrees = \sec \dfrac \pi 3 = 2$

Secant of $75 \degrees$

$\sec 75 \degrees = \sec \dfrac {5 \pi} {12} = \sqrt 6 + \sqrt 2$

Secant of Right Angle

$\sec 90 \degrees = \sec \dfrac \pi 2$ is undefined

Secant of $105 \degrees$

$\sec 105 \degrees = \sec \dfrac {7 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$

Secant of $120 \degrees$

$\sec 120 \degrees = \sec \dfrac {2 \pi} 3 = -2$

Secant of $135 \degrees$

$\sec 135 \degrees = \sec \dfrac {3 \pi} 4 = -\sqrt 2$

Secant of $150 \degrees$

$\sec 150 \degrees = \sec \dfrac {5 \pi} 6 = -\dfrac {2 \sqrt 3} 3$

Secant of $165 \degrees$

$\sec 165 \degrees = \sec \dfrac {11 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$

Secant of Straight Angle

$\sec 180 \degrees = \sec \pi = -1$

Secant of $195 \degrees$

$\sec 195 \degrees = \sec \dfrac {13 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$

Secant of $210 \degrees$

$\sec 210 \degrees = \sec \dfrac {7 \pi} 6 = -2 \dfrac {\sqrt 3} 3$

Secant of $225 \degrees$

$\sec 225 \degrees = \sec \dfrac {5 \pi} 4 = -\sqrt 2$

Secant of $240 \degrees$

$\sec 240 \degrees = \sec \dfrac {4 \pi} 3 = - 2$

Secant of $255 \degrees$

$\sec 255 \degrees = \sec \dfrac {17 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$

Secant of Three Right Angles

$\sec 270 \degrees = \sec \dfrac {3 \pi} 2$ is undefined

Secant of $285 \degrees$

$\sec 285 \degrees = \sec \dfrac {19 \pi} {12} = \sqrt 6 + \sqrt 2$

Secant of $300 \degrees$

$\sec 300 \degrees = \sec \dfrac {5 \pi} 3 = 2$

Secant of $315 \degrees$

$\sec 315 \degrees = \sec \dfrac {7 \pi} 4 = \sqrt 2$

Secant of $330 \degrees$

$\sec 330 \degrees = \sec \dfrac {11 \pi} 6 = 2 \dfrac {\sqrt 3} 3$

Secant of $345 \degrees$

$\sec 345 \degrees = \sec \dfrac {23 \pi} {12} = \sqrt 6 - \sqrt 2$

Secant of Full Angle

$\sec 360 \degrees = \sec 2 \pi = 1$

Also see

• Particular Values of Sine Function
• Particular Values of Cosine Function
• Particular Values of Tangent Function
• Particular Values of Cotangent Function
• Particular Values of Cosecant Function

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles