Particular Values of Sine Function

Theorem

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

This list is non-exhaustive.

Sine of Zero

$\sin 0 = 0$

Sine of 15 Degrees

$\sin 15 \degrees = \sin \dfrac \pi {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$

Sine of 30 Degrees

$\sin 30 \degrees = \sin \dfrac \pi 6 = \dfrac 1 2$

Sine of 45 Degrees

$\sin 45 \degrees = \sin \dfrac \pi 4 = \dfrac {\sqrt 2} 2$

Sine of 60 Degrees

$\sin 60 \degrees = \sin \dfrac \pi 3 = \dfrac {\sqrt 3} 2$

Sine of 75 Degrees

$\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$

Sine of Right Angle

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

Sine of 105 Degrees

$\sin 105^\circ = \sin \dfrac {7 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$

Sine of 120 Degrees

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

Sine of 135 Degrees

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

Sine of 150 Degrees

$\sin 150 \degrees = \sin \dfrac {5 \pi} 6 = \dfrac 1 2$

Sine of 165 Degrees

$\sin 165 \degrees = \sin \dfrac {11 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$

Sine of Straight Angle

$\sin 180 \degrees = \sin \pi = 0$

Sine of 195 Degrees

$\sin 195 \degrees = \sin \dfrac {13 \pi} {12} = -\dfrac {\sqrt 6 - \sqrt 2} 4$

Sine of 210 Degrees

$\sin 210 \degrees = \sin \dfrac {7 \pi} 6 = -\dfrac 1 2$

Sine of 225 Degrees

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

Sine of 240 Degrees

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

Sine of 255 Degrees

$\sin 255^\circ = \sin \dfrac {17 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$

Sine of Three Right Angles

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

Sine of 285 Degrees

$\sin 285^\circ = \sin \dfrac {19 \pi} {12} = - \dfrac {\sqrt 6 + \sqrt 2} 4$

Sine of 300 Degrees

$\sin 300 \degrees = \sin \dfrac {5 \pi} 3 = -\dfrac {\sqrt 3} 2$

Sine of 315 Degrees

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

Sine of 330 Degrees

$\sin 330 \degrees = \sin \dfrac {11 \pi} 6 = -\dfrac 1 2$

Sine of 345 Degrees

$\sin 345^\circ = \sin \dfrac {23 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$

Sine of Full Angle

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

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