Particular Values of Cosecant Function

Theorem

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

This list is non-exhaustive.

Cosecant of Zero

$\csc 0$ is undefined

Cosecant of 15 Degrees

$\csc 15 \degrees = \csc \dfrac \pi {12} = \sqrt 6 + \sqrt 2$

Cosecant of 30 Degrees

$\csc 30^\circ = \csc \dfrac \pi 6 = 2$

Cosecant of 45 Degrees

$\csc 45^\circ = \csc \dfrac \pi 4 = \sqrt 2$

Cosecant of 60 Degrees

$\csc 60^\circ = \csc \dfrac \pi 3 = \dfrac {2 \sqrt 3} 3$

Cosecant of 75 Degrees

$\csc 75 \degrees = \csc \dfrac {5 \pi} {12} = \sqrt 6 - \sqrt 2$

Cosecant of Right Angle

$\csc 90 \degrees = \csc \dfrac \pi 2 = 1$

Cosecant of 105 Degrees

$\csc 105 \degrees = \csc \dfrac {7 \pi} {12} = \sqrt 6 - \sqrt 2$

Cosecant of 120 Degrees

$\csc 120 \degrees = \csc \dfrac {2 \pi} 3 = \dfrac {2 \sqrt 3} 3$

Cosecant of 135 Degrees

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

Cosecant of 150 Degrees

$\csc 150 \degrees = \csc \dfrac {5 \pi} 6 = \sqrt 2$

Cosecant of 165 Degrees

$\csc 165 \degrees = \csc \dfrac {11 \pi} {12} = \sqrt 6 + \sqrt 2$

Cosecant of Straight Angle

$\csc 180 \degrees = \csc \pi$ is undefined

Cosecant of 195 Degrees

$\csc 195^\circ = \csc \dfrac {13 \pi} {12} = - \left({\sqrt 6 + \sqrt 2}\right)$

Cosecant of 210 Degrees

$\csc 210 \degrees = \csc \dfrac {7 \pi} 6 = -2$

Cosecant of 225 Degrees

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

Cosecant of 240 Degrees

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

Cosecant of 255 Degrees

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

Cosecant of Three Right Angles

$\csc 270 \degrees = \csc \dfrac {3 \pi} 2 = -1$

Cosecant of 285 Degrees

$\csc 285 \degrees = \csc \dfrac {19 \pi} {12} = -\paren {\sqrt 6 - \sqrt 2}$

Cosecant of 300 Degrees

$\csc 300^\circ = \csc \dfrac {5 \pi} 3 = -\dfrac {2 \sqrt 3} 3$

Cosecant of 315 Degrees

$\csc 315^\circ = \csc \dfrac {7 \pi} 4 = -\sqrt 2$

Cosecant of 330 Degrees

$\csc 330 \degrees = \csc \dfrac {11 \pi} 6 = -2$

Cosecant of 345 Degrees

$\csc 345 \degrees = \csc \dfrac {23 \pi} {12} = -\paren {\sqrt 6 + \sqrt 2}$

Cosecant of Full Angle

$\csc 360 \degrees = \csc 2 \pi$ is undefined

Also see

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