1.17 45° - Square

2 Notes

2.1 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.

2.2 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 - decagonA decagon is a polygon with ten sides and ten angles. The decagon is the largest polygon that exists in the Archimedean solids. The measure of each angle of a decagon is 144 degrees. See also geometry, decagonal number pentagon, hexagon, octagon Polygons. (10 sided)
    • 81°-9°-90° triangle - icosagonIn geometry, an icosagon is a twenty-sided polygon. An example: German Swastika is an irregular icosagon. The interior angle of a regular icosagon is 162°. Polygons. (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 - tricontagonA tricontagon is a polygon with 30 sides. The measure of each angle of a regular tricontagon is 168 degrees. Polygons. (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 - enneagonIn geometry, an enneagon or nonagon is a nine-sided polygon. The two different names are similarly derived: enneagon is derived from Greek enneagonon (εννεα, nine + γωνον, corner ; nonagon is borrowed (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)
    • ...

2.3 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 squareThe term perfect square is used in mathematics in two meanings: a positive integer which is the square of some other integer, i. can be written in the form n''2 for some integer n''. Examples: 1, 4, 9, 16, 25, 36, 49,. See square number. an algebraic expr

3 See also

• Trigonometric functionIn mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. They may be defined as ratios of two sides of a right triangle containing the angle, or, more generally, as ratios of
• Trigonometric identity

4 External links

• Constructible Polygons
• Constructible Regular Polygons
• Naming polygons

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