Particular Values of Cosine Function

Theorem

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

This list is non-exhaustive.

Cosine of Zero

$\cos 0 = 1$

Cosine of 15 Degrees

$\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$

Cosine of 30 Degrees

$\cos 30 \degrees = \cos \dfrac \pi 6 = \dfrac {\sqrt 3} 2$

Cosine of 45 Degrees

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

Cosine of 60 Degrees

$\cos 60 \degrees = \cos \dfrac \pi 3 = \dfrac 1 2$

Cosine of 75 Degrees

$\cos 75^\circ = \cos \dfrac {5 \pi}{12} = \dfrac {\sqrt 6 - \sqrt 2} 4$

Cosine of Right Angle

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

Cosine of 105 Degrees

$\cos 105 \degrees = \cos \dfrac {7 \pi} {12} = - \dfrac {\sqrt 6 - \sqrt 2} 4$

Cosine of 120 Degrees

$\cos 120 \degrees = \cos \dfrac {2 \pi} 3 = -\dfrac 1 2$

Cosine of 135 Degrees

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

Cosine of 150 Degrees

$\cos 150 \degrees = \cos \dfrac {5 \pi} 6 = -\dfrac {\sqrt 3} 2$

Cosine of 165 Degrees

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

Cosine of Straight Angle

$\cos 180 \degrees = \cos \pi = -1$

Cosine of 195 Degrees

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

Cosine of 210 Degrees

$\cos 210 \degrees = \cos \dfrac {7 \pi} 6 = -\dfrac {\sqrt 3} 2$

Cosine of 225 Degrees

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

Cosine of 240 Degrees

$\cos 240 \degrees = \cos \dfrac {4 \pi} 3 = -\dfrac 1 2$

Cosine of 255 Degrees

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

Cosine of Three Right Angles

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

Cosine of 285 Degrees

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

Cosine of 300 Degrees

$\cos 300 \degrees = \cos \dfrac {5 \pi} 3 = \dfrac 1 2$

Cosine of 315 Degrees

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

Cosine of 330 Degrees

$\cos 330 \degrees = \cos \dfrac {11 \pi} 6 = \dfrac {\sqrt 3} 2$

Cosine of 345 Degrees

$\cos 345 \degrees = \cos \dfrac {23 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$

Cosine of Full Angle

$\cos 360 \degrees = \cos 2 \pi = 1$

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