Exact trigonometric constants - AbsoluteAstronomy.com

Exact constant expressions for trigonometric values are sometimes useful, mainly for simplifying solutions into radical

Nth root

In mathematics, the nth root of a number x is a number r which, when raised to the power of n, equals xr^n = x,where n is the degree of the root...

forms which allow further simplification.

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 radian

Radian

Radian is the ratio between the length of an arc and its radius. The radian is the standard unit of angular measure, used in many areas of mathematics. The unit was formerly a SI supplementary unit, but this category was abolished in 1995 and the radian is now considered a SI derived unit...

Fermat number

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 the trigonometric functions of other angles, such as 2π/7, 2π/9 (= 40°), and 2π/13 (as well as the other constructible

Constructible number

A point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...

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
Generating trigonometric tables
In mathematics, tables of trigonometric functions are useful in a number of areas. Before the existence of pocket calculators, trigonometric tables were essential for navigation, science and engineering...

.

Table of constants

Values outside the [0°, 45°] angle range are trivially derived from these values, using circle axis reflection
Coordinate rotations and reflections
In geometry, 2D coordinate rotations and reflections are two kinds of Euclidean plane isometries which are related to one another.A rotation in the plane can be formed by composing a pair of reflections. First reflect a point P to its image P′ on the other side of line L1...

symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

. (See Trigonometric identity.)

0°: fundamental

3°: 60-sided polygon

6°: 30-sided polygon

9°: 20-sided polygon

12°: 15-sided polygon

15°: dodecagon

18°: decagon

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

60°: triangle

where

is the golden ratio

Golden ratio

In mathematics and the arts, two quantities are in the golden ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the smaller one. The golden ratio is an irrational mathematical constant, approximately 1.61803398874989...

Uses for constants

As an example of the use of these constants, consider a dodecahedron with the following volume, where a 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 constructibility

Constructibility

In mathematics, there are several notions of constructibility. Each of the following is by definition constructible:* a point in the Euclidean plane that can be constructed with compass and straightedge...

of right triangles.

Here 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. An n-gon can be divided into 2n right triangles with angles of {180/n, 90−180/n, 90} degrees, for n in 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
Constructible polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not....
- 3×2ⁿ-sided regular polygons, for n in 0, 1, 2, 3, ...
  - 30°-60°-90° triangle: triangle (3-sided)
  - 60°-30°-90° triangle: hexagon (6-sided)
  - 75°-15°-90° triangle: dodecagon
    Dodecagon
    In geometry, a dodecagon is any polygon with twelve sides and twelve angles.- Regular dodecagon :It usually refers to a regular dodecagon, having all sides of equal length and all angles equal to 150°...
    
    (12-sided)
  - 82.5°-7.5°-90° triangle: icosikaitetragon (24-sided)
  - 86.25°-3.75°-90° triangle: tetracontakaioctagon (48-sided)
  - ...
- 4×2ⁿ-sided
  - 45°-45°-90° triangle: square
    Square (geometry)
    In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...
    
    (4-sided)
  - 67.5°-22.5°-90° triangle: octagon (8-sided)
  - 78.75°-11.25°-90° triangle: hexakaidecagon
    Hexadecagon
    In mathematics, a hexadecagon is a polygon with 16 sides and 16 vertices.- Regular hexadecagon :A regular hexadecagon is constructible with a compass and straightedge....
    
    (16-sided)
  - ...
- 5×2ⁿ-sided
  - 54°-36°-90° triangle: pentagon
    Pentagon
    In geometry, a pentagon is any five-sided polygon. A pentagon may be simple or self-intersecting. The sum of the internal angles in a simple pentagon is 540°. A pentagram is an example of a self-intersecting pentagon.- Regular pentagons :In a regular pentagon, all sides are equal in length and...
    
    (5-sided)
  - 72°-18°-90° triangle: decagon
    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 each internal angle equal to 144°...
    
    (10-sided)
  - 81°-9°-90° triangle: icosagon
    Icosagon
    In geometry, an icosagon is a twenty-sided polygon. The sum of any icosagon's interior angles is 3240 degrees.One interior angle in a regular icosagon is 162° meaning that one exterior angle would be 18°...
    
    (20-sided)
  - 85.5°-4.5°-90° triangle: tetracontagon (40-sided)
  - 87.75°-2.25°-90° triangle: octacontagon (80-sided)
  - ...
- 15×2ⁿ-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...)
Nonconstructible (with whole or half degree angles) – No finite radical expressions involving real numbers for these triangle edge ratios are possible, therefore its multiples of two are also not possible.
- 9×2ⁿ-sided
  - 70°-20°-90° triangle: enneagon
    Enneagon
    In geometry, a nonagon is a nine-sided polygon.The name "nonagon" is a prefix hybrid formation, from Latin , used equivalently, attested already in the 16th century in French nonogone and in English from the 17th century...
    
    (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)
  - ...
- 45×2ⁿ-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)
  - ...

The trivial ones

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

n × π/(5 × 2^m)

Geometrical method

Applying Ptolemy's theorem

Ptolemy's theorem

In Euclidean geometry, Ptolemy's theorem is a relation between the four sides and two diagonals of a cyclic quadrilateral . The theorem is named after the Greek astronomer and mathematician Ptolemy...

to the cyclic quadrilateral

Cyclic quadrilateral

In Euclidean geometry, a cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle. This circle is called the circumcircle or circumscribed circle, and the vertices are said to be concyclic. Other names for these quadrilaterals are chordal quadrilateral and inscribed...

ABCD defined by four successive vertices of the pentagon, we can find that:

which is the reciprocal 1/φ of the golden ratio

Golden ratio

. Crd is the Chord function,

Thus

(Alternatively, without using Ptolemy's theorem, label as X the intersection of AC and BD, and note by considering angles that triangle AXB is isosceles, so AX = AB = a. Triangles AXD and CXB are similar, because AD is parallel to BC. So XC = a·(a/b). But AX + XC = AC, so a + a²/b = b. Solving this gives a/b = 1/φ, as above).

Similarly

Algebraic method

The multiple angle formulas for functions of

, where

and

, can be solved for the functions of

, since we know the function values of

. The multiple angle formulas are:

When or , we let or and solve for :

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

When or , we again let or and solve for :

which factors into:

n × π/20

9° is 45-36, and 27° is 45−18; so we use the subtraction formulas for sine and cosine.

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 sine and cosine.

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 sine and cosine.

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

In general nested radicals cannot be reduced.

But if for

is rational, and both

are rational, with the appropriate choice of the four ± signs, then

For example,

External links

The source of this article is wikipedia, the free encyclopedia. The text of this article is licensed under the GFDL.