Exact trigonometric constants Details, Meaning Exact trigonometric constants Article and Explanation Guide

Exact trigonometric constants Guide, Meaning , Facts, Information and Description

Exact constant expressions for trigonometric 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, Double-angle, Addition/subtraction and values for 0°, 30°, 36° and 45°. Note that 1° = π/180 radians.

Table of constants

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

0° Fundamental

15° - 12-sided polygon

18° - 10-sided polygon

21° - Sum 9 + 12

sin(21°) = [2√(5-√5)(√3+1)-√2(√3-1)(1+√5)]/16

cos(21°) = [2√(5-√5)(√3-1)+√2(√3+1)(1+√5)]/16

tan(21°) = [√(5-2√5)(1+2√3-√5)+(2+√3)(√5-3)+2]/2

22.5° - Octagon

24° - Sum 12° + 12°

sin(24°) = √(2(5+√5))(1-√5)+2√3(1+√5))/16

cos(24°) = √(6(5+√5))(√5-1)+2(1+√5))/16

tan(24°) = (√(10+2√5)-2√3)(3+√5)/4

cotan(24°) = (√(10+2√5)+2√3)(√5-1)/4

27° - Sum 12° + 15°

sin(27°) = ((2√(5+√5)+√2(1-√5))/8

cos(27°) = ((2√(5+√5)+√2(√5-1))/8

tan(27°) = -√(5-2(√5))+(√5-1)

30° - Hexagon

33° - Sum 15° + 18°

sin(33°) = (2√(5+√5)(-1+√3)+√2(√5-1)(1+√3))/16

cos(33°) = (2√(5+√5)(+1+√3)+√2(√5-1)(1-√3))/16

tan(33°) = (√(5(5-2√5))(-15+10√3-7√5+4√15)+5((-2+√3)(3+√5)+2))/10

36° - Pentagon

39° - Sum 18°+ 21°

sin(39°) = (2√(5-√5)(1-√3)+√2(+1+√3)(1+√5))/16

cos(39°) = (2√(5-√5)(1+√3)+√2(-1+√3)(1+√5))/16

tan(39°) = (√(2(5+√5))-2)((2-√3)(-3+√5)+2)/4

42° - Sum 21° + 21°

sin(42°) = (√(6(5-√5))(1+√5)+2(1-√5))/16

cos(42°) = (√(2(5-√5))(1+√5)+2√3(-1+√5))/16

tan(42°) = (-√(5-2√5)(3+√5)+√3(1+√5))/2

45° - Square

Notes

Uses for constants

As an example of the use of these constants, the volume of a dodecahedron is

V = 5e³cos(36°)/tan²(36°)

Using

cos(36°) = (√5 + 1)/4

tan(36°) = √(5-2√5)

this can be be simplified to:

V = e³(15 + 7√5)/4.

Derivation triangles

The derivation of sin, 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 3 points in a regular polygon: A vertex, an edge center containing that vertex, and the polygon center. A N-agon can be divided into 2*N right triangle with angles of {180/N, 90-180/N, 90} degrees, where 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.

Constructable
- 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 constructable regular polygons don't make whole degree angles: 17, 51, 85, 255, 257...)
Nonconstructable (with whole or half degree angles) - No finite radial forms for these triangle edge ratios are known.
- 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)
  - ...

Expressions not unique

Simplifying nested radical expressions is nontrivial. The expressions here may not all be fully reduced.

Example:

It's not evident that this simplification is equivalent, and in general nested radials can not be reduced.

In general this is reducible:

, if (a²-4*b²*c) is a perfect square

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