Editing Trigonometric functions (section)

==Algebraic values==
[[File:Unit circle angles color.svg|right|thumb|The [[unit circle]], with some points labeled with their cosine and sine (in this order), and the corresponding angles in radians and degrees.]]
The [[algebraic expression]]s for the most important angles are as follows:

:<math>\sin 0 = \sin 0^\circ \quad= \frac{\sqrt0}2 = 0</math> ([[Angle#Types of angles|zero angle]])
:<math>\sin \frac\pi6 = \sin 30^\circ = \frac{\sqrt1}2 = \frac{1}{2}</math>
:<math>\sin \frac\pi4 = \sin 45^\circ = \frac{\sqrt{2}}{2} = \frac{1}{\sqrt{2}}</math>
:<math>\sin \frac\pi3 = \sin 60^\circ = \frac{\sqrt{3}}{2}</math>
:<math>\sin \frac\pi2 = \sin 90^\circ = \frac{\sqrt4}2 = 1</math> ([[right angle]])

Writing the numerators as [[square roots]] of consecutive non-negative integers, with a denominator of 2, provides an easy way to remember the values.<ref name="Larson_2013"/>

Such simple expressions generally do not exist for other angles which are rational multiples of a right angle.
*For an angle which, measured in degrees, is a multiple of three, the [[exact trigonometric values]] of the sine and the cosine may be expressed in terms of square roots. These values of the sine and the cosine may thus be constructed by [[Compass-and-straightedge construction|ruler and compass]].
*For an angle of an integer number of degrees, the sine and the cosine may be expressed in terms of square roots and the [[cube root]] of a non-real [[complex number]]. [[Galois theory]] allows a proof that, if the angle is not a multiple of 3°, non-real cube roots are unavoidable.
*For an angle which, expressed in degrees, is a [[rational number]], the sine and the cosine are [[algebraic number]]s, which may be expressed in terms of [[nth root|{{mvar|n}}th roots]]. This results from the fact that the [[Galois group]]s of the [[cyclotomic polynomial]]s are [[cyclic group|cyclic]].
*For an angle which, expressed in degrees, is not a rational number, then either the angle or both the sine and the cosine are [[transcendental number]]s. This is a corollary of [[Baker's theorem]], proved in 1966.
*If the sine of an angle is a rational number then the cosine is not necessarily a rational number, and vice-versa.  However if the tangent of an angle is rational then both the sine and cosine of the double angle will be rational.

===Simple algebraic values===
{{main|Exact trigonometric values#Common angles}}

The following table lists the sines, cosines, and tangents of multiples of 15 degrees from 0 to 90 degrees.
 
{| class="wikitable" style="text-align:center;"
|-
! colspan=2 | Angle, ''θ'', in
! rowspan=2 | <math>\sin(\theta)</math>
! rowspan=2 | <math>\cos(\theta)</math>
! rowspan=2 | <math>\tan(\theta)</math>
|-
! radians
! degrees
|-
| <math>0</math>
| <math>0^\circ</math>
| <math>0</math>
| <math>1</math>
| <math>0</math>
|-
| <math>\frac{\pi}{12}</math>
| <math>15^\circ</math>
| <math>\frac{\sqrt{6}-\sqrt{2}}{4}</math>
| <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| <math>2-\sqrt{3}</math>
|-
| <math>\frac{\pi}{6}</math>
| <math>30^\circ</math>
| <math>\frac{1}{2}</math>
| <math>\frac{\sqrt{3}}{2}</math>
| <math>\frac{\sqrt{3}}{3}</math>
|-
| <math>\frac{\pi}{4}</math>
| <math>45^\circ</math>
| <math>\frac{\sqrt{2}}{2}</math>
| <math>\frac{\sqrt{2}}{2}</math>
| <math>1</math>
|-
| <math>\frac{\pi}{3}</math>
| <math>60^\circ</math>
| <math>\frac{\sqrt{3}}{2}</math>
| <math>\frac{1}{2}</math>
| <math>\sqrt{3}</math>
|-
| <math>\frac{5\pi}{12}</math>
| <math>75^\circ</math>
| <math>\frac{\sqrt{6}+\sqrt{2}}{4}</math>
| <math>\frac{\sqrt{6}-\sqrt{2}}{4}</math>
| <math>2 + \sqrt{3}</math>
|-
| <math>\frac{\pi}{2}</math>
| <math>90^\circ</math>
| <math>1</math>
| <math>0</math>
| {{n/a|Undefined}}
|}