Particular Values of Cosecant Function
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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
- Particular Values of Sine Function
- Particular Values of Cosine Function
- Particular Values of Tangent Function
- Particular Values of Cotangent Function
- Particular Values of Secant 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