Exact trigonometric values

In mathematics, the values of the trigonometric functions can be expressed approximately, as in Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos (\pi/4) \approx 0.707 , or exactly, as in Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos (\pi/ 4)= \sqrt 2 /2 . While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.

Common angles

The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 90°. In the table below, the label "Undefined" represents a ratio Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 1:0.

If the codomain of the trigonometric functions is taken to be the real numbers these entries are undefined, whereas if the codomain is taken to be the projectively extended real numbers, these entries take the value $\infty$ (see division by zero).

Radians	Degrees	sin	cos	tan	cot	sec	csc
$0$	$0^\circ$	$0$	$1$	$0$	Undefined	$1$	Undefined
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\pi}{12}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 15^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt6 - \sqrt2} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt6 + \sqrt2} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 - \sqrt3	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 + \sqrt3	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt6 - \sqrt2	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt6 + \sqrt2
$\frac{\pi}{10}$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 18^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt5 - 1} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{10 + 2\sqrt5}} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{25 - 10\sqrt5}} {5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{5 + 2\sqrt5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{50 - 10\sqrt5}} {5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt5 + 1
$\frac{\pi}{8}$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 22.5^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{2 - \sqrt2}} {2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{2 + \sqrt2}} {2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt2 - 1	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt2 + 1	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{4 - 2\sqrt2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{4 + 2\sqrt2}
$\frac{\pi}{6}$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 30^\circ	$\frac{1}{2}$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt3}{2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt3}{3}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt3	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{2\sqrt3}{3}	$2$
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\pi}{5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 36^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{10 - 2\sqrt5}} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt5 + 1} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{5 - 2\sqrt5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{25 + 10\sqrt5}} {5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt5 - 1	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{50 + 10\sqrt5}} {5}
$\frac{\pi}{4}$	$45^\circ$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt2}{2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt2}{2}	$1$	$1$	$\sqrt2$	$\sqrt2$
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{3\pi}{10}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 54^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt5 + 1}{4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{10 - 2\sqrt5}} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{25 + 10\sqrt5}} {5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{5 - 2\sqrt5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{50 + 10\sqrt5}} {5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt5 - 1
$\frac{\pi}{3}$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 60^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt3}{2}	$\frac{1}{2}$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt3	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt3}{3}	$2$	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{2\sqrt3}{3}
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{3\pi}{8}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 67.5^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{2 + \sqrt2}} {2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{2 - \sqrt2}} {2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt2 + 1	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt2 - 1	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{4 + 2\sqrt2}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{4 - 2\sqrt2}
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{2\pi}{5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 72^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{10 + 2\sqrt5}} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt5 - 1}{4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{5 + 2\sqrt5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{25 - 10\sqrt5}} {5}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt5 + 1	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt{50 - 10\sqrt5}} {5}
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{5\pi}{12}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 75^\circ	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt6 + \sqrt2} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{\sqrt6 - \sqrt2} {4}	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 + \sqrt3	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 - \sqrt3	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt6 + \sqrt2	Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt6 - \sqrt2
$\frac{\pi}{2}$	$90^\circ$	$1$	$0$	Undefined	$0$	Undefined	$1$

For angles outside of this range, trigonometric values can be found by applying reflection and shift identities such as

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{alignat}{3} &&\sin(2\pi + \theta) &{}= \sin(\pi - \theta) &&{}= \sin(\theta), \quad &&\sin(\pi + \theta) &&{}= \sin(-\theta) &&{}= -\sin(\theta), \\[5mu] &&\cos(2\pi + \theta) &{}= \cos(-\theta) &&{}= \cos(\theta), \quad &&\cos(\pi + \theta) &&{}= \cos(\pi - \theta) &&{}= -\cos(\theta). \end{alignat}

Trigonometric numbers

A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of π radians. Since Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(x)=\cos(x-\pi/2),

the case of a sine can be omitted from this definition. Therefore any trigonometric number can be written as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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 Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \left(\cos\left(\frac{2\pi k}{n}\right) + i \sin\left(\frac{2\pi k}{n}\right)\right)^n = \cos(2\pi k) + i \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 minimal polynomials of trigonometric numbers can be explicitly enumerated. In contrast, by the Lindemann–Weierstrass theorem, the sine or cosine of any non-zero algebraic number is always transcendental.

The real part of any root of unity is a trigonometric number. By Niven's theorem, the only rational trigonometric numbers are 0, 1, −1, 1/2, and −1/2.

Constructibility

An angle can be constructed with a compass and straightedge if and only if its sine (or equivalently cosine) can be expressed by a combination of arithmetic operations and square roots applied to integers. Additionally, an angle that is a rational multiple of $\pi$ radians is constructible if and only if, when it is expressed as Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): a\pi/b

radians, where a and b are relatively prime integers, the prime factorization of the denominator, b, is the product of some power of two and any number of distinct Fermat primes (a Fermat prime is a prime number one greater than a power of two).

Thus, for example, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2\pi/15 = 24^\circ

is a constructible angle because 15 is the product of the Fermat primes 3 and 5. Similarly Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \pi/12 = 15^\circ
is a constructible angle because 12 is a power of two (4) times a Fermat prime (3). But Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \pi/9 = 20^\circ
is not a constructible angle, since Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 9 = 3 \cdot 3
is not the product of distinct Fermat primes as it contains 3 as a factor twice, and neither is Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \pi/7 \approx 25.714^\circ

, since 7 is not a Fermat prime.

It results from the above characterisation that an angle of an integer number of degrees is constructible if and only if this number of degrees is a multiple of 3.

Constructible values

45°

From a reflection identity, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos(45^\circ) = \sin(90^\circ-45^\circ)=\sin(45^\circ) . Substituting into the Pythagorean trigonometric identity Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(45^\circ)^2 + \cos(45^\circ)^2=1 , one obtains the minimal polynomial Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2 \sin(45^\circ)^2 - 1 = 0 . Taking the positive root, one finds Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(45^\circ) = \cos(45^\circ) = 1/\sqrt{2} = \sqrt{2}/2 .

30° and 60°

The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle with angles 30-60-90 is obtained. By symmetry, the bisected side is half of the side of the equilateral triangle, so one concludes Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(30^\circ)=1/2 . The Pythagorean and reflection identities then give Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(60^\circ)=\cos(30^\circ)=\sqrt{1-(1/2)^2}=\sqrt{3}/2 .

18°, 36°, 54°, and 72°

The value of Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(18^\circ)

may be derived using the multiple angle formulas for sine and cosine. By the double angle formula for sine:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(36^\circ) = 2\sin(18^\circ)\cos(18^\circ)

By the triple angle formula for cosine:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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°):

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 2\sin(18^\circ) = 1 - 4\sin^2(18^\circ)

This quadratic equation has only one positive root:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(18^\circ) = \frac{\sqrt{5}-1}{4}

The Pythagorean identity then gives Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos(18^\circ) , and the double and triple angle formulas give sine and cosine of 36°, 54°, and 72°.

Remaining multiples of 3°

The sines and cosines of all other angles between 0 and 90° that are multiples of 3° can be derived from the angles described above and the sum and difference formulas. Specifically,

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} 3^\circ &= 18^\circ-15^\circ, & 24^\circ &= 54^\circ - 30^\circ, & 51^\circ &= 60^\circ - 9^\circ, & 78^\circ &= 60^\circ + 18^\circ, & \\ 6^\circ &= 36^\circ - 30^\circ, & 27^\circ &= 45^\circ - 18^\circ, & 57^\circ &= 30^\circ + 27^\circ, & 81^\circ &= 45^\circ + 36^\circ, & \\ 9^\circ &= 45^\circ - 36^\circ, & 33^\circ &= 60^\circ - 27^\circ, & 63^\circ &= 45^\circ + 18^\circ, & 84^\circ &= 54^\circ + 30^\circ, & \\ 12^\circ &= 30^\circ - 18^\circ, & 39^\circ &= 30^\circ + 9^\circ, & 66^\circ &= 36^\circ + 30^\circ, & 87^\circ &= 60^\circ + 27^\circ. & \\ 15^\circ &= 45^\circ - 30^\circ, & 42^\circ &= 60^\circ - 18^\circ, & 69^\circ &= 60^\circ + 9^\circ, & \\ 21^\circ &= 30^\circ - 9^\circ, & 48^\circ &= 30^\circ + 18^\circ, & 75^\circ &= 45^\circ + 30^\circ, & \end{align}

For example, since Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 24^\circ = 60^\circ - 36^\circ , its cosine can be derived by the cosine difference formula:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align}\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{align}

Half angles

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:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(22.5^\circ) = \sqrt{\frac{1 - \cos(45^\circ)}{2}} = \sqrt{\frac{1 - \frac{\sqrt{2}}{2}}{2}} = \frac12\sqrt{2-\sqrt{2}}

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \cos(22.5^\circ) = \sqrt{\frac{1 + \cos(45^\circ)}{2}} = \sqrt{\frac{1 + \frac{\sqrt{2}}{2}}{2}} = \frac12\sqrt{2+\sqrt{2}}

Repeated application of the half-angle formulas leads to nested radicals, specifically nested square roots of 2 of the form Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sqrt{2 \pm \cdots} . In general, the sine and cosine of most angles of the form Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \beta / 2^n

can be expressed using nested square roots of 2 in terms of $\beta$. Specifically, if one can write an angle as

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {\displaystyle \alpha = \pi \left(\frac{1}{2}-\sum_{i=1}^k \frac{\prod_{j=1}^i b_j}{2^{i+1}}\right) = \pi \left (\frac{1}{2} - \frac{b_1}{4} - \frac{b_1 b_2}{8} - \frac{b_1 b_2 b_3}{16} - \ldots - \frac{b_1 b_2 \ldots b_k}{2^{k+1}}\right)}

where Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b_k \in [-2,2]

and $b_i$ is -1, 0, or 1 for Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): i<k

, then Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {\displaystyle \cos(\alpha) = \frac{b_1}{2}\sqrt{2+b_2 \sqrt{2+b_3 \sqrt{2+\ldots+b_{k-1} \sqrt{2+2 \sin\left(\frac{\pi b_k}{4}\right)}}}} }

and if Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): b_1 \neq 0

then

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {\displaystyle \sin(\alpha) = \frac{1}{2}\sqrt{2 - b_2 \sqrt{2+b_3 \sqrt{2+b_4 \sqrt{2+\ldots+b_{k-1} \sqrt{2+2 \sin\left(\frac{\pi b_k}{4}\right)}}}}} }

For example, Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{13 \pi}{32} = \pi\left(\frac{1}{2}-\frac{1}{4}+\frac{1}{8}+\frac{1}{16}-\frac{1}{32}\right) , so one has Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): (b_1,b_2,b_3,b_4)=(1,-1,1,-1)

and obtains:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {\displaystyle \cos\left(\frac{13 \pi}{32}\right) = \frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2+2 \sin\left(\frac{-\pi}{4}\right)}}} = \frac{1}{2}\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}}

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): {\displaystyle \sin\left(\frac{13 \pi}{32}\right) = \frac{1}{2}\sqrt{2+\sqrt{2+\sqrt{2-\sqrt 2}}}}

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:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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 of the form Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{k 2^n \pi}{17}

(for integers Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): k,n

) can be derived from this one.

Non-constructibility of 1°

As discussed in #Constructibility § Notes, only certain angles that are rational multiples of $\pi$ radians have trigonometric values that can be expressed with square roots. The angle 1°, being Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \pi/180 = \pi/(2^2 \cdot 3^2 \cdot 5)

radians, has a repeated factor of 3 in the denominator and therefore Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(1^\circ)
cannot be expressed using only square roots. A related question is whether it can be expressed using cube roots. The following two approaches can be used, but both result in an expression that involves the cube root of a complex number.

Using the triple-angle identity, we can identify Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(1^\circ)

as a root of a cubic polynomial: Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(3^\circ) = -4x^3 + 3x

. The three roots of this polynomial are Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(1^\circ) , Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(59^\circ) , and Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): -\sin(61^\circ) . Since Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(3^\circ)

is constructible, an expression for it could be plugged into Cardano's formula to yield an expression for Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \sin(1^\circ)

. However, since all three roots of the cubic are real, this is an instance of casus irreducibilis, and the expression would require taking the cube root of a complex number.

Alternatively, by De Moivre's formula:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} (\cos(1^\circ) + i \sin(1^\circ))^3 &= \cos(3^\circ) + i \sin(3^\circ), \\[4mu] (\cos(1^\circ) - i \sin(1^\circ))^3 &= \cos(3^\circ) - i \sin(3^\circ). \end{align}

Taking cube roots and adding or subtracting the equations, we have:

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \begin{align} \cos(1^\circ) &= \;\frac{1}{2} \left( \sqrt[3]{\cos(3^\circ) + i \sin(3^\circ)} + \sqrt[3]{\cos(3^\circ) - i \sin(3^\circ)} \right), \\[5mu] \sin(1^\circ) &= \frac{1}{2i}\left( \sqrt[3]{\cos(3^\circ) + i \sin(3^\circ)} - \sqrt[3]{\cos(3^\circ) - i \sin(3^\circ)} \right). \end{align}

See also

• Ailles rectangle, used to find exact values for multiples of 15°
• List of trigonometric identities