Particular Values of Tangent Function

Theorem

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

This list is non-exhaustive.

Tangent of Zero

$\tan 0 = 0$

Tangent of 15 Degrees

$\tan 15^\circ = \tan \dfrac {\pi} {12} = 2 - \sqrt 3$

Tangent of 22.5 Degrees

$\tan 22.5 \degrees = \tan \dfrac \pi 8 = \sqrt 2 - 1$

Tangent of 30 Degrees

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

Tangent of 45 Degrees

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

Tangent of 60 Degrees

$\tan 60 \degrees = \tan \dfrac \pi 3 = \sqrt 3$

Tangent of 67.5 Degrees

$\tan 67.5 \degrees = \tan \dfrac {3 \pi} 8 = \sqrt 2 + 1$

Tangent of 75 Degrees

$\tan 75 \degrees = \tan \dfrac {5 \pi} {12} = 2 + \sqrt 3$

Tangent of Right Angle

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

Tangent of 105 Degrees

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

Tangent of 120 Degrees

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

Tangent of 135 Degrees

$\tan 135^\circ = \tan \dfrac {3 \pi} 4 = - 1$

Tangent of 150 Degrees

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

Tangent of 165 Degrees

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

Tangent of Straight Angle

$\tan 180 \degrees = \tan \pi = 0$

Tangent of 195 Degrees

$\tan 195 \degrees = \tan \dfrac {13 \pi} {12} = 2 - \sqrt 3$

Tangent of 210 Degrees

$\tan 210 \degrees = \tan \dfrac {7 \pi} 6 = \dfrac {\sqrt 3} 3$

Tangent of 225 Degrees

$\tan 225 \degrees = \tan \dfrac {5 \pi} 4 = 1$

Tangent of 240 Degrees

$\tan 240 \degrees = \tan \dfrac {4 \pi} 3 = \sqrt 3$

Tangent of 255 Degrees

$\tan 255 \degrees = \tan \dfrac {17 \pi} {12} = 2 + \sqrt 3$

Tangent of Three Right Angles

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

Tangent of 285 Degrees

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

Tangent of 300 Degrees

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

Tangent of 315 Degrees

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

Tangent of 330 Degrees

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

Tangent of 345 Degrees

$\tan 345^\circ = \tan \dfrac {23 \pi} {12} = -\left({2 - \sqrt 3}\right)$

Tangent of Full Angle

$\tan 360^\circ = \tan 2 \pi = 0$

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