Exact trigonometric values

In mathematics, a trigonometric number is an irrational number that can be expressed as the sine or cosine of an angle that is a rational multiple of π radians. Such an angle would be a rational number of degrees or turns. For example, since $\cos(\pi /4)={\sqrt {2}}/2,$ this number is trigonometric.

Common angles

Main article: Trigonometric_functions § Simple_algebraic_values

The following table lists the sine and cosine of angles from 0 to 45 degrees that are multiples of 15, 18, or 22.5 degrees. The values of sine and cosine that are irrational are trigonometric numbers. The sines and cosines of angles from 45 to 90 degrees can be inferred from the identity cos(x) = sin(90° - x).

x in radians	x in degrees	$\sin(x)$	$\cos(x)$
$0$	$0^{\circ }$	$0$	$1$
${\frac {\pi }{12}}$	$15^{\circ }$	${\frac {{\sqrt {6}}-{\sqrt {2}}}{4}}$	${\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}$
${\frac {\pi }{10}}$	$18^{\circ }$	${\frac {{\sqrt {5}}-1}{4}}$	${\frac {\sqrt {10+2{\sqrt {5}}}}{4}}$
${\frac {\pi }{8}}$	$22.5^{\circ }$	${\frac {\sqrt {2-{\sqrt {2}}}}{2}}$	${\frac {\sqrt {2+{\sqrt {2}}}}{2}}$
${\frac {\pi }{6}}$	$30^{\circ }$	${\frac {1}{2}}$	${\frac {\sqrt {3}}{2}}$
${\frac {\pi }{5}}$	$36^{\circ }$	${\frac {\sqrt {10-2{\sqrt {5}}}}{4}}$	${\frac {{\sqrt {5}}+1}{4}}$
${\frac {\pi }{4}}$	$45^{\circ }$	${\frac {\sqrt {2}}{2}}$	${\frac {\sqrt {2}}{2}}$

The derivation of these values is discussed below.

Real parts of roots of unity

Any trigonometric number can be written as $\cos(2\pi k/n)$, where k and n are integers. This number can be thought of as the real part of the complex number $\cos(2\pi k/n)+i\sin(2\pi k/n)$. De Moivre's formula shows that numbers of this form are roots of unity:

$\left(\cos \left({\frac {2\pi k}{n}}\right)+i\sin \left({\frac {2\pi k}{n}}\right)\right)^{n}=\cos(2\pi k)+\sin(2\pi k)=1$

Since the root of unity is a root of the polynomial xⁿ - 1, it is algebraic. Since the trigonometric number is the average of the root of unity and its complex conjugate, and algebraic numbers are closed under arithmetic operations, every trigonometric number is algebraic.

The real part of any root of unity is trigonometric, unless it is rational. By Niven's theorem, the only rational numbers that can be expressed as the real part of a root of unity are 0, 1, -1, 1/2, and -1/2.

Expressibility with square roots

Some trigonometric numbers, such as ${\sqrt {3}}/2$, can be expressed in terms of a combination of arithmetic operations and square roots. Such numbers are called constructible, because one length can be constructed by compass and straightedge from another if and only if the ratio between the two lengths is such a number. However, some trigonometric numbers, such as $\cos(20^{\circ })$, have been proven to not be constructible.

A trigonometric number $\sin(\theta )$ or $\cos(\theta )$ is constructible if and only if the angle $\theta$ is constructible. Let $\theta =q\pi$, where q = a/b and a and b are relatively prime integers. Then the angle is constructible if and only if the prime factorization of the denominator, b, consists of any number of Fermat primes, each with an exponent of 1, times any power of two. For example, $15^{\circ }$ and $24^{\circ }$ are constructible because they are equivalent to $\pi /12$ and $2\pi /15$ radians, respectively, and 12 is the product of 3 and 4, which are a Fermat prime and a power of two, and 15 is the product of Fermat primes 3 and 5. On the other hand, $20^{\circ }$ is not constructible because it corresponds to a denominator of 9 = 3², and the Fermat prime cannot be raised to a power. As another example, $(360/7)^{\circ }$ is not constructible, because the denominator of 7 is not a Fermat prime.

Derivations of constructible values

The values of trigonometric numbers can be derived through a combination of methods. The values of sine and cosine of 30, 45, and 60 degrees are derived by analysis of the 30-60-90 and 90-45-45 triangles. If the angle is expressed in radians as $a\pi /b$, this takes care of the case where a is 1 and b is 2, 3, 4, or 6.

Half-angle formula

If the denominator, b, is multiplied by additional factors of 2, the sine and cosine can be derived with the half-angle formulas. For example, 22.5° (π/8 rad) is half of 45°, so its sine and cosine are:

$\sin(22.5^{\circ })={\sqrt {\frac {1-\cos(45^{\circ })}{2}}}={\sqrt {\frac {1-{\frac {\sqrt {2}}{2}}}{2}}}={\frac {\sqrt {2-{\sqrt {2}}}}{2}}$

$\cos(22.5^{\circ })={\sqrt {\frac {1+\cos(45^{\circ })}{2}}}={\sqrt {\frac {1+{\frac {\sqrt {2}}{2}}}{2}}}={\frac {\sqrt {2+{\sqrt {2}}}}{2}}$

Repeated application of the cosine half-angle formula leads to nested square roots that continue in a pattern where each application adds a ${\sqrt {2+\cdots }}$ to the numerator and the denominator is 2. For example:

$\cos \left({\frac {\pi }{16}}\right)={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\qquad \cos \left({\frac {\pi }{32}}\right)={\frac {\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}{2}}$

$\cos \left({\frac {\pi }{12}}\right)={\frac {{\sqrt {6}}+{\sqrt {2}}}{4}}={\frac {\sqrt {2+{\sqrt {3}}}}{2}}\qquad \cos \left({\frac {\pi }{24}}\right)={\frac {\sqrt {2+{\sqrt {2+{\sqrt {3}}}}}}{2}}$

Sine of 18°

Cases where the denominator, b, is 5 times a power of 2 can start from the following derivation of $\sin(18^{\circ })$, since $18^{\circ }=\pi /10$ radians. The derivation uses the multiple angle formulas for sine and cosine. By the double angle formula for sine:

$\sin(36^{\circ })=2\sin(18^{\circ })\cos(18^{\circ })$

By the triple angle formula for cosine:

$\cos(54^{\circ })=\cos ^{3}(18^{\circ })-3\sin ^{2}(18^{\circ })\cos(18^{\circ })=\cos(18^{\circ })(1-4\sin ^{2}(18^{\circ }))$

Since sin(36°) = cos(54°), we equate these two expressions and cancel a factor of cos(18°):

$2\sin(18^{\circ })=1-4\sin ^{2}(18^{\circ })$

This quadratic equation has only one positive root:

$\sin(18^{\circ })={\frac {{\sqrt {5}}-1}{4}}$

Using other identities

The sines and cosines of many other angles can be derived using the results described above and a combination of the multiple angle formulas and the sum and difference formulas. For example, for the case where b is 15 times a power of 2, since $24^{\circ }=60^{\circ }-36^{\circ }$, its cosine can be derived by the cosine difference formula:

${\begin{aligned}\cos(24^{\circ })&=\cos(60^{\circ })\cos(36^{\circ })+\sin(60^{\circ })\sin(36^{\circ })\\&={\frac {1}{2}}{\frac {{\sqrt {5}}+1}{4}}+{\frac {\sqrt {3}}{2}}{\frac {\sqrt {10-2{\sqrt {5}}}}{4}}\\&={\frac {1+{\sqrt {5}}+{\sqrt {30-6{\sqrt {5}}}}}{8}}\end{aligned}}$

Denominator of 17

Main article: Heptadecagon

Since 17 is a Fermat prime, a regular 17-gon is constructible, which means that the sines and cosines of angles such as $2\pi /17$ radians can be expressed in terms of square roots. In particular, in 1796, Carl Friedrich Gauss showed that:

$\cos \left({\frac {2\pi }{17}}\right)={\frac {-1+{\sqrt {17}}+{\sqrt {34-2{\sqrt {17}}}}+2{\sqrt {17+3{\sqrt {17}}-{\sqrt {170+38{\sqrt {17}}}}}}}{16}}$

The sines and cosines of other constructible angles with a denominator divisible by 17 can be derived from this one.