NationMaster - Encyclopedia: Exact trigonometric constants

Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. Trigonometry (from the Greek trigonon = three angles and metro = measure) is a branch of mathematics dealing with angles, triangles and trigonometric functions such as sine and cosine. ... In mathematics, an nth root of a number a is a number b, such that bn=a. ...

All values of sine, cosine, and tangent of angles with 3° increments are derivable using identities: Half-angle, Double-angle, Addition/subtraction and values for 0°, 30°, 36° and 45°. Note that 1° = π/180 radians. In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ... Some common angles, measured in radians. ...

This article is incomplete in at least two senses. First, it is always possible to apply a half-angle formula and find an exact expression for the cosine of one-half the smallest angle on the list. Second, this article exploits only the first two of five known Fermat primes: 3 and 5; and tbe trigonometric functions of other angles, such as 2π/7, 2π/9 (= 40°), and 2π/13 (as well as the other constructible polygons, 2π/17, 2π/257, or 2π/65537) are soluble by radicals. In practice, all values of sine, cosine, and tangent not found in this article are approximated using the techniques described at Generating trigonometric tables. In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the form where n is a nonnegative integer. ... A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), one can construct the point with unruled straightedge and compass. ... In mathematics, more specifically in abstract algebra, Galois theory, named after Ã‰variste Galois, provides a connection between field theory and group theory. ... Tables of trigonometric functions are useful in a number of areas. ...

1 Table of constants
2 Notes
- 2.1 Uses for constants
- 2.2 Derivation triangles
3 How can the trig values for sin and cos be calculated?
4 How can the trig values for tan and cot be calculated?
5 Plans for simplifying
6 See also
7 External links

Table of constants

Values outside [0°,45°] angle range are trivially extracted from circle axis reflection symmetry from these values. (See Trigonometric identity) In geometry, coordinate rotations and reflections are two kinds of isometry which are related to each to other. ... Sphere symmetry group o. ... In mathematics, trigonometric identities are equations involving trigonometric functions that are true for all values of the occurring variables. ...

0° Fundamental

$sin 0^circ = 0$

$cos 0^circ = 1$

$tan 0^circ = 0$

$cot 0^circ = No Solution$

3° - 60-sided polygon

$sin frac {pi}{60} = sin 3^circ = frac{ 2 (1 - sqrt3) sqrt{5 + sqrt5} + sqrt2 (sqrt5 - 1) (sqrt3 + 1) }{16}$

$cos frac {pi}{60} = cos 3^circ = frac{ 2 (1 + sqrt3) sqrt{5 + sqrt5} + sqrt2 (sqrt5 - 1) (sqrt3 - 1) }{16}$

$tan frac {pi}{60} = tan 3^circ = frac{ left( (2 - sqrt3) (3 + sqrt5) - 2 right) left(2 - sqrt{2 (5 - sqrt5)}right) }{4}$

$cot frac {pi}{60} = cot 3^circ = frac{ left( (2 + sqrt3) (3 + sqrt5) - 2 right) left(2 + sqrt{2 (5 - sqrt5)}right) }{4}$

6° - 30-sided polygon

$sin frac {pi}{30} = sin 6^circ = frac{(sqrt6) sqrt{5 - sqrt5} - (sqrt5 + 1)}{8}$

$cos frac {pi}{30} = cos 6^circ = frac{(sqrt2) sqrt{5- sqrt5} + sqrt3( sqrt5+1)}{8}$

$tan frac {pi}{30} = tan 6^circ = frac{(sqrt2) sqrt{5 - sqrt5} - sqrt3(sqrt5 - 1)}{2}$

$cot frac {pi}{30} = cot 6^circ = frac{sqrt3 (3 + sqrt5) + sqrt{51 + 22 sqrt5}}{2}$

9° - 20-sided polygon

$sin frac {pi}{20} = sin 9^circ = frac{sqrt2(sqrt5 + 1) - 2sqrt{5 - sqrt5}}{8}$

$cos frac {pi}{20} = cos 9^circ = frac{sqrt2(sqrt5 + 1) + 2sqrt{5 - sqrt5}}{8}$

$tan frac {pi}{20} = tan 9^circ = sqrt5 + 1 - sqrt{5 + 2sqrt5}$

$cot frac {pi}{20} = cot 9^circ = sqrt5 + 1 + sqrt{5 + 2sqrt5}$

12° - 15-sided polygon

$sin frac{pi}{15} = sin 12^circ = frac{(sqrt2) sqrt{5 + sqrt5} - sqrt 3 (sqrt 5 -1)}{8}$

$cos frac{pi}{15} = cos 12^circ = frac{(sqrt6) sqrt{5 + sqrt5} + (sqrt 5 - 1)}{8}$

$tan frac{pi}{15} = tan 12^circ = frac{(sqrt3) (3 - sqrt5 ) - sqrt{50 - 22 sqrt5}}{2}$

$cot frac{pi}{15} = cot 12^circ = frac{sqrt3 (sqrt5 + 1) + sqrt2 sqrt{5 + sqrt5}}{2}$

15° - 12-sided polygon

$sin frac{pi}{12} = sin 15^circ = frac{sqrt 2 left(sqrt 3 - 1right)}{4}$

$cos frac{pi}{12} = cos 15^circ = frac{sqrt 2 left(sqrt 3 + 1right)}{4}$

$tan frac{pi}{12} = tan 15^circ = 2 - sqrt 3$

$cot frac{pi}{12} = cot 15^circ = 2 + sqrt 3$

18° - 10-sided polygon

$sin frac{pi}{10} = sin 18^circ = frac{sqrt 5 - 1}{4}$

$cos frac{pi}{10} = cos 18^circ = frac{sqrt{2(5 + sqrt 5)}}{4}$

$tan frac{pi}{10} = tan 18^circ = frac{sqrt{5(5 - 2 sqrt 5)}}{5}$

$cot frac{pi}{10} = cot 18^circ = sqrt{5 + 2 sqrt 5}$

21° - Sum 9° + 12°

$sin frac{7pi}{60} = sin 21^circ = frac{2(sqrt 3 + 1) sqrt{5 - sqrt 5} - sqrt 2 (sqrt 3 - 1) (1 + sqrt 5)} {16}$

$cos frac{7pi}{60} = cos 21^circ = frac{2 (sqrt 3 - 1) sqrt{5 - sqrt 5} + sqrt 2 (sqrt 3 + 1) (1 + sqrt 5)} {16}$

$tan frac {7 pi}{60} = tan 21^circ = frac{ left(2 - (2 + sqrt3) (3 - sqrt5) right) left(2 - sqrt{2 (5 + sqrt5)}right) }{4}$

$cot frac {7 pi}{60} = cot 21^circ = frac{ left(2 - (2 - sqrt3) (3 - sqrt5) right) left(2 + sqrt{2 (5 +sqrt5)}right) }{4}$

22.5° - Octagon

$sin frac {pi}{8} = sin 22.5^circ = frac{sqrt{2 - sqrt{2}}}{2}$

$cos frac {pi}{8} = cos 22.5^circ = frac{sqrt{2 + sqrt{2}}}{2}$

$tan frac {pi}{8} = tan 22.5^circ = sqrt{2}-1$

$cot frac {pi}{8} = cot 22.5^circ = sqrt{2}+1$

24° - Sum 12° + 12°

$sin frac {2pi}{15} = sin 24^circ = frac{sqrt3(sqrt5 + 1) - sqrt2 sqrt{5 - sqrt5}}{8}$

$cos frac {2pi}{15} = cos 24^circ = frac{sqrt6 sqrt{5 - sqrt5} + sqrt5 + 1}{8}$

$tan frac {2pi}{15} = tan 24^circ = frac{sqrt{50 + 22 sqrt5} - sqrt3 (3 + sqrt5)}{2}$

$cot frac {2pi}{15} = cot 24^circ = frac{sqrt2 sqrt{5 - sqrt5} + sqrt3(sqrt5 - 1)}{2}$

27° - Sum 12° + 15°

$sin frac {3pi}{20} = sin 27^circ = frac{2 sqrt{5 + sqrt5} - sqrt2(sqrt5 - 1)}{8}$

$cos frac {3pi}{20} = cos 27^circ = frac{2 sqrt{5 + sqrt5} + sqrt2(sqrt5 - 1)}{8}$

$tan frac {3pi}{20} = tan 27^circ = sqrt5 - 1 - sqrt{5 - 2 sqrt5}$

$cot frac {3pi}{20} = cot 27^circ = sqrt5 - 1 + sqrt{5 - 2 sqrt5}$

30° - Hexagon

$sin frac{pi}{6} = sin 30^circ = frac{1}{2}$

$cos frac{pi}{6} = cos 30^circ = frac{sqrt 3}{2}$

$tan frac{pi}{6} = tan 30^circ = frac{sqrt 3}{3}$

$cot frac{pi}{6} = cot 30^circ = frac{3}{sqrt 3} = sqrt 3$

33° - Sum 15° + 18°

$sin frac{11pi}{60} = sin 33^circ = frac{2(sqrt 3 - 1) sqrt{5 + sqrt 5} + sqrt 2 (1 + sqrt 3) (sqrt 5 - 1)} {16}$

$cos frac{11pi}{60} = cos 33^circ = frac{2 (sqrt 3 + 1) sqrt{5 + sqrt 5} + sqrt 2 (1 - sqrt 3) (sqrt 5 - 1)} {16}$

$tan frac {11 pi}{60} = tan 33^circ = frac{ left(2 - (2 - sqrt3) (3 + sqrt5) right) left(2 + sqrt{2 (5 - sqrt5)}right) }{4}$

$cot frac {11 pi}{60} = cot 33^circ = frac{ left(2 - (2 + sqrt3) (3 + sqrt5) right) left(2 - sqrt{2 (5 - sqrt5)}right) }{4}$

36° - Pentagon

$sin frac{pi}{5} = sin 36^circ = frac{sqrt{2(5 - sqrt 5)} }{4}$

$cos frac{pi}{5} = cos 36^circ = frac{sqrt 5+1}{4}$

$tan frac{pi}{5} = tan 36^circ = sqrt{5 - 2sqrt 5}$

$cot frac{pi}{5} = cot 36^circ = frac{ sqrt{5(5 + 2sqrt 5)}}{5}$

39° - Sum 18°+ 21°

$sin{frac{13pi}{60}} = sin{39^circ} = frac{2(1-sqrt 3)sqrt{5-sqrt 5} + sqrt 2 (sqrt 3 + 1)(sqrt 5 + 1)}{16}$

$cos{frac{13pi}{60}} = cos{39^circ} = frac{2 (1+sqrt 3)sqrt{5-sqrt 5} + sqrt2(sqrt 3 - 1)(sqrt 5 + 1)}{16}$

$tan frac {13pi}{60} = tan 39^circ = frac{ left( (2 - sqrt3) (3 - sqrt5) - 2 right) left(2 - sqrt{2 (5 + sqrt5)}right) }{4}$

$cot frac {13pi}{60} = cot 39^circ = frac{ left( (2 + sqrt3) (3 - sqrt5) - 2 right) left(2 + sqrt{2 (5 + sqrt5)}right) }{4}$

42° - Sum 21° + 21°

$sin frac {7pi}{30} = sin 42^circ = frac{ sqrt6 sqrt{5 + sqrt5} - (sqrt5 + 1)}{8}$

$cos frac {7pi}{30} = cos 42^circ = frac{ sqrt2 sqrt{5 + sqrt5} + sqrt3(sqrt5 - 1)}{8}$

$tan frac {7pi}{30} = tan 42^circ = frac{ sqrt3(sqrt5 + 1) - sqrt2 sqrt{5 + sqrt5}}{2}$

$cot frac {7pi}{30} = cot 42^circ = frac{ sqrt{50 - 22 sqrt5} + sqrt3(3 - sqrt5)}{2}$

45° - Square

$sin frac{pi}{4} = sin 45^circ = frac{sqrt 2}{2}$

$cos frac{pi}{4} = cos 45^circ = frac{sqrt 2}{2}$

$tan frac{pi}{4} = tan 45^circ = 1$

$cot frac{pi}{4} = cot 45^circ = 1$

60° - Triangle

$sin frac{pi}{3} = sin 60^circ = frac{sqrt 3}{2}$

$cos frac{pi}{3} = cos 60^circ = frac{1}{2}$

$tan frac{pi}{3} = tan 60^circ = sqrt 3$

$cot frac{pi}{3} = cot 60^circ = frac{sqrt 3}{3}$

Notes

Uses for constants

As an example of the use of these constants, consider a dodecahedron with the following volume, where e is the length of an edge: A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ...

$V = frac{5e^3cos{36^circ}}{tan^2{36^circ}}$

Using

$cos{36^circ} = frac{sqrt 5 + 1}{4}$

$tan{36^circ} = sqrt{5 - 2 sqrt 5}$

this can be simplified to:

$V = frac{ e^3left(15 + 7 sqrt 5 right)}{4}$

Derivation triangles

Regular polygon (N-sided) and its fundamental right triangle. Angle: a = 180/N °

The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructability of right triangles. Example polygon and its fundamental right triangle File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... Example polygon and its fundamental right triangle File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ...

Here are right triangles made from symmetry sections of regular polygons are used to calculate fundamental trigonometric ratios. Each right triangle represents three points in a regular polygon: a vertex, an edge center containing that vertex, and the polygon center. A N-gon can be divided into 2N right triangle with angles of {180/N, 90−180/N, 90} degrees, for N = 3, 4, 5, ...

Constructibility of 3, 4, 5, and 15 sided polygons are the basis, and angle bisectors allow multiples of two to also be derived.

Constructible
- 3×2^X-sided regular polygons, X = 0, 1, 2, 3, ...
  - 30°-60°-90° triangle - triangle (3 sided)
  - 60°-30°-90° triangle - hexagon (6-sided)
  - 75°-15°-90° triangle - dodecagon (12-sided)
  - 82.5°-7.5°-90° triangle - icosikaitetragon (24-sided)
  - 86.25°-3.75°-90° triangle - tetracontakaioctagon (48-sided)
  - ...
- 4×2^X-sided
  - 45°-45°-90° triangle - square (4-sided)
  - 67.5°-22.5°-90° triangle - octagon (8-sided)
  - 88.75°-11.25°-90° triangle - hexakaidecagon (16-sided)
  - ...
- 5×2^X-sided
  - 54°-36°-90° triangle - pentagon (5-sided)
  - 72°-18°-90° triangle - decagon (10-sided)
  - 81°-9°-90° triangle - icosagon (20-sided)
  - 85.5°-4.5°-90° triangle - tetracontagon (40-sided)
  - 87.75°-2.25°-90° triangle - octacontagon (80-sided)
  - ...
- 15×2^X-sided
  - 78°-12°-90° triangle - pentakaidecagon (15-sided)
  - 84°-6°-90° triangle - tricontagon (30-sided)
  - 87°-3°-90° triangle - hexacontagon (60-sided)
  - 88.5°-1.5°-90° triangle - hectoicosagon (120-sided)
  - 89.25°-0.75°-90° triangle - dihectotetracontagon (240-sided)
- ... (Higher constructible regular polygons don't make whole degree angles: 17, 51, 85, 255, 257...)

Nonconstructable (with whole or half degree angles) - No finite radical expressions involving real numbers for these triangle edge ratios are possible because of Casus Irreducibilis.
- 9×2^X-sided
  - 70°-20°-90° triangle - enneagon (9-sided)
  - 80°-10°-90° triangle - octakaidecagon (18-sided)
  - 85°-5°-90° triangle - triacontakaihexagon (36-sided)
  - 87.5°-2.5°-90° triangle - heptacontakaidigon (72-sided)
  - ...
- 45×2^X-sided
  - 86°-4°-90° triangle - tetracontakaipentagon (45-sided)
  - 88°-2°-90° triangle - enneacontagon (90-sided)
  - 89°-1°-90° triangle - hectaoctacontagon (180-sided)
  - 89.5°-0.5°-90° triangle - trihectohexacontagon (360-sided)
  - ...

In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. ... For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... A regular hexagon. ... In geometry, a dodecagon is a polygon with exactly twelve sides. ... For other uses, see Square. ... For other uses, see Octagon (disambiguation). ... In mathematics, a Hexadecagon (sometimes called a hexakaidecagon) is a polygon with 16 sides and 16 angles. ... Look up pentagon in Wiktionary, the free dictionary. ... a regular decagon In geometry, a decagon is any polygon with ten sides and ten angles, and usually refers to a regular decagon, having all sides of equal length and all angles equal to 144Â°, therefore making each angle of a regular decagon be 144Â°. Its SchlÃ¤fli symbol is... A regular icosagon. ... A tetracontagon is a polygon with 40 sides and 40 vertexes. ... Look up Polygon in Wiktionary, the free dictionary. ... A regular pentadecagon. ... Image:Triacontagon. ... For other uses, see Polygon (disambiguation). ... A regular enneagon. ... An octadecagon is a polygon with 18 sides and 18 vertexes. ...

How can the trig values for sin and cos be calculated?

The trivial ones

In degree format: 0, 90, 45, 30 and 60 can be calculated from their triangles, using the Pythagorean theorem.

n π / 10

The multiple angle formulas for functions of 5x, where x = {18, 36, 54, 72, 90} and 5x = {90, 180, 270, 360, 540}, can be solved for the functions of x, since we know the function values of 5x. The multiple angle formulas are:

sin5 x = 16sin 5 x - 20sin 3 x + 5sin x

cos5 x = 16cos 5 x - 20cos 3 x + 5cos x

When sin 5x = 0 or cos 5x = 0, we let y = sin x or y = cos x and solve for y:

16 y 5 - 20 y 3 + 5 y = 0

One solution is zero, and the resulting 4th degree equation can be solved as a quadratic in y-squared.

When sin 5x = 1 or cos 5x = 1, we again let y = sin x or y = cos x and solve for y:

16 y 5 - 20 y 3 + 5 y - 1 = 0

which factors into

(y - 1)(4 y 2 + 2 y - 1) 2 = 0

n π / 20

9° is 45 - 36, and 27° is 45 - 18; so we use the subtraction formulas for sin and cos.

n π / 30

6° is 36 - 30, 12° is 30 - 18, 24° is 54 - 30, and 42° is 60 - 18; so we use the subtraction formulas for sin and cos.

n π / 60

3° is 18 - 15, 21° is 36 - 15, 33° is 18 + 15, and 39° is 54 - 15, so we use the subtraction (or addition) formulas for sin and cos.

How can the trig values for tan and cot be calculated?

Tangent is sine divided by cosine, and cotangent is cosine divided by sine, or 1 divided by tangent.

Set up each fraction and simplify.

Plans for simplifying

Rationalize the denominator

If the denominator is a square root, multiply the numerator and denominator by that radical.

If the denominator is the sum or difference of two terms, multiply the numerator and denominator by the conjugate of the denominator. The conjugate is the identical, except the sign between the terms is changed.

Sometimes you need to rationalize the denominator more than once.

Split a fraction in two

Sometimes it helps to split the fraction into the sum of two fractions and then simplify both separately.

Squaring and square rooting

If there is a complicated term, with only one kind of radical in a term, this plan may help. Square the term, combine like terms, and take the square root. This may leave a big radical with a smaller radical inside, but it is often better than the original.

Simplification of nested radical expressions

Main article: Nested radical

In general nested radicals cannot be reduced. In algebra, nested radicals are radical expressions that have another radical expression nested inside a radical. ...

But if for $sqrt{a + bsqrt c}$ ,

$R = sqrt{a^2 - b^2 c}$ is rational,

and both $d = pm sqrt{ frac{ a pm R }{2}}$ and $e = pm sqrt{ frac{ a pm R }{2c}}$ are rational,

with the appropriate choice of the four $pm$ signs,

then $sqrt{a + bsqrt c} = d + esqrt c$

Example:

$4sin 18^circ = sqrt{6 - 2 sqrt 5} = sqrt 5 - 1$

External links

Constructible Polygons
Constructible Regular Polygons
Naming polygons

Categories: Trigonometry | Algebraic numbers

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