Particular Values of Cotangent Function

Theorem

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

This list is non-exhaustive.

Cotangent of Zero

$\cot 0$ is undefined

Cotangent of 15 Degrees

$\cot 15 \degrees = \cot \dfrac {\pi} {12} = 2 + \sqrt 3$

Cotangent of 30 Degrees

$\cot 30 \degrees = \cot \dfrac \pi 6 = \sqrt 3$

Cotangent of 45 Degrees

$\cot 45 \degrees = \cot \dfrac \pi 4 = 1$

Cotangent of 60 Degrees

$\cot 60 \degrees = \cot \dfrac \pi 3 = \dfrac {\sqrt 3} 3$

Cotangent of 75 Degrees

$\cot 75 \degrees = \cot \dfrac {5 \pi} {12} = 2 - \sqrt 3$

Cotangent of Right Angle

$\cot 90 \degrees = \cot \dfrac \pi 2 = 0$

Cotangent of 105 Degrees

$\cot 105^\circ = \cot \dfrac {7 \pi} {12} = -\left({2 - \sqrt 3}\right)$

Cotangent of 120 Degrees

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

Cotangent of 135 Degrees

$\cot 135 \degrees = \cot \dfrac {3 \pi} 4 = -1$

Cotangent of 150 Degrees

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

Cotangent of 165 Degrees

$\cot 165 \degrees = \cot \dfrac {11 \pi} {12} = -\paren {2 + \sqrt 3}$

Cotangent of Straight Angle

$\cot 180^\circ = \cot \pi$ is undefined

Cotangent of 195 Degrees

$\cot 195 \degrees = \cot \dfrac {13 \pi} {12} = 2 + \sqrt 3$

Cotangent of 210 Degrees

$\cot 210^\circ = \cot \dfrac {7 \pi} 6 = \sqrt 3$

Cotangent of 225 Degrees

$\cot 225^\circ = \cot \dfrac {5 \pi} 4 = 1$

Cotangent of 240 Degrees

$\cot 240^\circ = \cot \dfrac {4 \pi} 3 = \dfrac {\sqrt 3} 3$

Cotangent of 255 Degrees

$\cot 255 \degrees = \cot \dfrac {17 \pi} {12} = 2 - \sqrt 3$

Cotangent of Three Right Angles

$\cot 270 \degrees = \cot \dfrac {3 \pi} 2 = 0$

Cotangent of 285 Degrees

$\cot 285 \degrees = \cot \dfrac {19 \pi} {12} = -\paren {2 - \sqrt 3}$

Cotangent of 300 Degrees

$\cot 300 \degrees = \cot \dfrac {5 \pi} 3 = - \dfrac {\sqrt 3} 3$

Cotangent of 315 Degrees

$\cot 315 \degrees = \cot \dfrac {7 \pi} 4 = -1$

Cotangent of 330 Degrees

$\cot 330^\circ = \cot \dfrac {11 \pi} 6 = -\sqrt 3$

Cotangent of 345 Degrees

$\cot 345 \degrees = \cot \dfrac {23 \pi} {12} = -\paren {2 + \sqrt 3}$

Cotangent of Full Angle

$\cot 360^\circ = \cot 2 \pi$ 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