Exact trigonometric constants
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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: Halfangle, Doubleangle, Addition/subtraction and values for 0°, 30°, 36°, and 45°. Note that 1° = radians.
This article is incomplete in at least two senses. First, it is always possible to apply a halfangle formula and find an exact expression for the cosine of 1/2 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 (= 40°), and (as well as the other constructible polygons, , or 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.
The values of sine, cosine, and tangent of angles with 1° increments can also be derived using the tripleangle identity. However, they can only be expressed with intermediate (and irreductible) complex numbers in the expression (see the formulae to compute the roots of a cubic equation), or by transforming them using hyperbolic real functions).
Table of constants
Values outside [0°, 45°] angle range are trivially extracted from circle axis reflection symmetry from these values. (See Trigonometric identity)
0°: fundamental
3°: 60sided polygon
6°: 30sided polygon
9°: 20sided polygon
12°: 15sided 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
Notes
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 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 ngon can be divided into 2n right triangle with angles of { 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
 3×2^{n}sided regular polygons, for n in 0, 1, 2, 3, ...
 30°60°90° triangle: triangle (3 sided)
 60°30°90° triangle: hexagon (6sided)
 75°15°90° triangle: dodecagon (12sided)
 82.5°7.5°90° triangle: icosikaitetragon (24sided)
 86.25°3.75°90° triangle: tetracontakaioctagon (48sided)
 ...
 4×2^{n}sided
 45°45°90° triangle: square (4sided)
 67.5°22.5°90° triangle: octagon (8sided)
 78.75°11.25°90° triangle: hexakaidecagon (16sided)
 ...
 5×2^{n}sided
 54°36°90° triangle: pentagon (5sided)
 72°18°90° triangle: decagon (10sided)
 81°9°90° triangle: icosagon (20sided)
 85.5°4.5°90° triangle: tetracontagon (40sided)
 87.75°2.25°90° triangle: octacontagon (80sided)
 ...
 15×2^{n}sided
 78°12°90° triangle: pentakaidecagon (15sided)
 84°6°90° triangle: tricontagon (30sided)
 87°3°90° triangle: hexacontagon (60sided)
 88.5°1.5°90° triangle: hectoicosagon (120sided)
 89.25°0.75°90° triangle: dihectotetracontagon (240sided)
 ... (Higher constructible regular polygons don't make whole degree angles: 17, 51, 85, 255, 257...)
 3×2^{n}sided regular polygons, for n in 0, 1, 2, 3, ...
 Nonconstructable (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.
 9×2^{n}sided
 70°20°90° triangle: enneagon (9sided)
 80°10°90° triangle: octakaidecagon (18sided)
 85°5°90° triangle: triacontakaihexagon (36sided)
 87.5°2.5°90° triangle: heptacontakaidigon (72sided)
 ...
 45×2^{n}sided
 86°4°90° triangle: tetracontakaipentagon (45sided)
 88°2°90° triangle: enneacontagon (90sided)
 89°1°90° triangle: hectaoctacontagon (180sided)
 89.5°0.5°90° triangle: trihectohexacontagon (360sided)
 ...
 9×2^{n}sided
Calculated trigonometric values for sine and cosine
The trivial ones
In degree format: 0, 90, 45, 30 and 60 can be calculated from their triangles, using the Pythagorean theorem.
n × π/(5×2^{m})
Geometrical method
Applying Ptolemy's theorem to the cyclic quadrilateral ABCD defined by four successive vertices of the pentagon, we can find that:
which is the reciprocal 1/f of the 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+a^{2}/b=b. Solving this gives a/b=1/f, as above).
Similarly
so
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 4536, and 27° is 45−18; so we use the subtraction formulas for sine and cosine.
n × π/30
 6° is 3630, 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.
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
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:
See also
 Trigonometric function
 Trigonometric identity
 Constructible polygon
 Trigonometric number
 17gonal construction
References
 Weisstein, Eric W., "Constructible polygon" from MathWorld.
 Weisstein, Eric W., "Trigonometry angles" from MathWorld.
 π/3 (60°) — π/6 (30°) — π/12 (15°) — π/24 (7.5°)
 π/4 (45°) — π/8 (22.5°) — π/16 (11.25°) — π/32 (6.625°)
 π/5 (36°) — π/10 (18°) — π/20 (9°)
 π/7 — π/14
 π/9 (20°) — π/18 (10°)
 π/11
 π/13
 π/15 (12°) — π/30 (6°)
 π/17
 π/19
 π/23
External links
ca:Constants trigonomètriques exactes
eo:Akurataj trigonometriaj konstantojhe:קבועים טריגונומטריים מדויקים
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