Exact trigonometric constants

Exact constant expressions for trigonometric Trigonometry

Trigonometry is a branch of mathematics [i] dealing with angle [i]s, triangle [i]s and trigonometric function [i] ...

expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification. All values of sine, cosine, and tangent of angles with 3 increments are derivable using identities: Half-angle List of trigonometric identities

In mathematics [i], trigonometric identities are equalities involving trigonometric function [i]s that a ...

, Double-angle List of trigonometric identities

In mathematics [i], trigonometric identities are equalities involving trigonometric function [i]s that a ...

, Addition/subtraction List of trigonometric identities

In mathematics [i], trigonometric identities are equalities involving trigonometric function [i]s that a ...

and values for 0, 30, 36 and 45. Note that 1 = p/180 radian Radian

The radian is a unit of plane angle [i]. ...

s. This article is incomplete in at least two senses.

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Exact constant expressions for trigonometric Trigonometry

Trigonometry is a branch of mathematics [i] dealing with angle [i]s, triangle [i]s and trigonometric function [i] ...

expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification.

All values of sine, cosine, and tangent of angles with 3° increments are derivable using identities: Half-angle List of trigonometric identities

In mathematics [i], trigonometric identities are equalities involving trigonometric function [i]s that a ...

, Double-angle List of trigonometric identities

In mathematics [i], trigonometric identities are equalities involving trigonometric function [i]s that a ...

, Addition/subtraction List of trigonometric identities

In mathematics [i], trigonometric identities are equalities involving trigonometric function [i]s that a ...

and values for 0°, 30°, 36° and 45°. Note that 1° = p/180 radian Radian

The radian is a unit of plane angle [i]. ...

s.

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. One could in principle write down formulae involving the angles 2p/17, 2p/257, or 2p/65537, but they would be too unwieldy for most applications. In practice, all values of sine, cosine, and tangent not found in this article are approximated using the techniques described at Generating trigonometric tables.

Table of constants

Values outside 0° ... 45° angle range are trivially extracted from circle axis reflection symmetry Symmetry

Symmetry is a characteristic feature of geometrical [i] shapes, system [i]s, equation [i]s, and ...

from these values.

0° Fundamental

3° - 60-sided polygon

6° - 30-sided polygon

9° - 20-sided polygon

12° - 15-sided polygon

15° - 12-sided polygon

18° - 10-sided polygon

21° - Sum 9° + 12°

22.5° - Octagon

24° - Sum 12° + 12°

27° - Sum 12° + 15°

30° - Hexagon

33° - Sum 15° + 18°

36° - Pentagon

39° - Sum 18°+ 21°

42° - Sum 21° + 21°

45° - Square

Notes

Uses for constants

As an example of the use of these constants, consider a dodecahedron Dodecahedron

A dodecahedron is any polyhedron [i] with twelve faces, but usually a regular dodecahedron is mean ...

with the following volume, where e is the length of an edge:

Using

this can be simplified to:

Derivation triangles

The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructability of right triangles.

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 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 Triangle

A triangle is one of the basic shape [i]s of geometry [i]: a polygon [i] with three vertices [i] ...
60°-30°-90° triangle - hexagon Hexagon

In geometry [i], a hexagon is a polygon [i] with six edge [i]s and six vertices [i]. ...
75°-15°-90° triangle - dodecagon Dodecagon

In geometry [i], a dodecagon is a polygon [i] with exactly twelve sides. ...
82.5°-7.5°-90° triangle - icosikaitetragon
86.25°-3.75°-90° triangle - tetracontakaioctagon
...

4×2^X-sided

45°-45°-90° triangle - square
67.5°-22.5°-90° triangle - octagon Octagon

In geometry [i], an octagon is a polygon [i] that has eight [i] sides.
...

- - 88.75°-11.25°-90° triangle - hexakaidecagon
  - ...
- 5×2^X-sided

...

- - 72°-18°-90° triangle - decagon Decagon
    
    In geometry [i], a decagon is any polygon [i] with ten sides and ten angle [i]s, and usually refers to a ...
  - 81°-9°-90° triangle - icosagon Icosagon
    
    In geometry [i], an icosagon is a twenty-sided polygon [i].

...

- - 85.5°-4.5°-90° triangle - tetracontagon
  - 87.75°-2.25°-90° triangle - octacontagon Polygon
    
    A polygon is a closed [i] planar [i] path composed of a finite number of sequential ...
  - ...
- 15×2^X-sided
- ...

Nonconstructable - 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 Enneagon

In geometry [i], an enneagon or nonagon is a nine [i]-sided polygon [i]. ...
80°-10°-90° triangle - octakaidecagon
85°-5°-90° triangle - triacontakaihexagon
87.5°-2.5°-90° triangle - heptacontakaidigon
...

45×2^X-sided

86°-4°-90° triangle - tetracontakaipentagon
88°-2°-90° triangle - enneacontagon
89°-1°-90° triangle - hectaoctacontagon
89.5°-0.5°-90° triangle - trihectohexacontagon
...

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 p over 10

The multiple angle formulas for functions of 5x, where x = and 5x = , can be solved for the functions of x, since we know the function values of 5x. The multiple angle formulas are:

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

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:

which factors into

n p over 20

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

n p over 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 p over 60

3° is 18 - 15, 21° is 36 - 15, 33° is 18 + 15, and 39° is 54 - 15, so we use the subtraction 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.

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 radicals

In general nested radicals cannot be reduced.

But if for ,

is rational,

and both
and are rational,

with the appropriate choice of the four signs,

then

Example: