Trigonometric constants expressed in real radicals

  (Redirected from Exact trigonometric constants)
The primary solution angles in the form (cos,sin) on the unit circle are at multiples of 30 and 45 degrees.

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

All trigonometric numbers – sines or cosines of rational multiples of 360° – are algebraic numbers (solutions of polynomial equations with integer coefficients); moreover they may be expressed in terms of radicals of complex numbers; but not all of these are expressible in terms of real radicals. When they are, they are expressible more specifically in terms of square roots.

All values of the sines, cosines, and tangents of angles at 3° increments are expressible in terms of square roots, using identities – the half-angle identity, the double-angle identity, and the angle addition/subtraction identity – and using values for 0°, 30°, 36°, and 45°. For an angle of an integer number of degrees that is not a multiple 3° (π/60 radians), the values of sine, cosine, and tangent cannot be expressed in terms of real radicals.

According to Niven's theorem, the only rational values of the sine function for which the argument is a rational number of degrees are 0, 1/2,  1, −1/2, and −1.

According to Baker's theorem, if the value of a sine, a cosine or a tangent is algebraic, then the angle is either a rational number of degrees or a transcendental number of degrees. That is, if the angle is an algebraic, but non-rational, number of degrees, the trigonometric functions all have transcendental values.

Scope of this articleEdit

The list in this article is incomplete in several senses. First, the trigonometric functions of all angles that are integer multiples of those given can also be expressed in radicals, but some are omitted here.

Second, it is always possible to apply the half-angle formula to find an expression in radicals for a trigonometric function of one-half of any angle on the list, then half of that angle, etc.

Third, expressions in real radicals exist for a trigonometric function of a rational multiple of π if and only if the denominator of the fully reduced rational multiple is a power of 2 by itself or the product of a power of 2 with the product of distinct Fermat primes, of which the known ones are 3, 5, 17, 257, and 65537.

Fourth, this article only deals with trigonometric function values when the expression in radicals is in real radicals – roots of real numbers. Many other trigonometric function values are expressible in, for example, cube roots of complex numbers that cannot be rewritten in terms of roots of real numbers. For example, the trigonometric function values of any angle that is one-third of an angle θ considered in this article can be expressed in cube roots and square roots by using the cubic equation formula to solve

 

but in general the solution for the cosine of the one-third angle involves the cube root of a complex number (giving casus irreducibilis).

In practice, all values of sines, cosines, and tangents not found in this article are approximated using the techniques described at Trigonometric tables.

Table of some common anglesEdit

Several different units of angle measure are widely used, including degrees, radians, and gradians (gons):

1 full circle (turn) = 360 degrees = 2π radians  =  400 gons.

The following table shows the conversions and values for some common angles:

Turns Degrees Radians Gradians sine cosine tangent
0 0 0g 0 1 0
1/12 30° π/6 33+1/3g 1/2 3/2 3/3
1/8 45° π/4 50g 2/2 2/2 1
1/6 60° π/3 66+2/3g 3/2 1/2 3
1/4 90° π/2 100g 1 0
1/3 120° 2π/3 133+1/3g 3/2 1/2 3
3/8 135° 3π/4 150g 2/2 2/2 −1
5/12 150° 5π/6 166+2/3g 1/2 3/2 3/3
1/2 180° π 200g 0 −1 0
7/12 210° 7π/6 233+1/3g 1/2 3/2 3/3
5/8 225° 5π/4 250g 2/2 2/2 1
2/3 240° 4π/3 266+2/3g 3/2 1/2 3
3/4 270° 3π/2 300g −1 0
5/6 300° 5π/3 333+1/3g 3/2 1/2 3
7/8 315° 7π/4 350g 2/2 2/2 −1
11/12 330° 11π/6 366+2/3g 1/2 3/2 3/3
1 360° 2π 400g 0 1 0

Further anglesEdit

 
Exact trigonometric table for multiples of 3 degrees.

Values outside the [0°, 45°] angle range are trivially derived from these values, using circle axis reflection symmetry. (See List of trigonometric identities.)

In the entries below, when a certain number of degrees is related to a regular polygon, the relation is that the number of degrees in each angle of the polygon is (n – 2) times the indicated number of degrees (where n is the number of sides). This is because the sum of the angles of any n-gon is 180° × (n – 2) and so the measure of each angle of any regular n-gon is 180° × (n – 2) ÷ n. Thus for example the entry "45°: square" means that, with n = 4, 180° ÷ n = 45°, and the number of degrees in each angle of a square is (n – 2) × 45° = 90°.

0°: fundamentalEdit

 
 
 
 

1.5°: regular hecatonicosagon (120-sided polygon)Edit

 
 

1.875°: regular enneacontahexagon (96-sided polygon)Edit

 
 

2.25°: regular octacontagon (80-sided polygon)Edit

 
 

2.8125°: regular hexacontatetragon (64-sided polygon)Edit

 
 

3°: regular hexacontagon (60-sided polygon)Edit

 
 
 
 

3.75°: regular tetracontaoctagon (48-sided polygon)Edit

 
 

4.5°: regular tetracontagon (40-sided polygon)Edit

 
 

5.625°: regular triacontadigon (32-sided polygon)Edit

 
 

6°: regular triacontagon (30-sided polygon)Edit

 
 
 
 

7.5°: regular icositetragon (24-sided polygon)Edit

 
 
 
 

9°: regular icosagon (20-sided polygon)Edit

 
 
 
 

11.25°: regular hexadecagon (16-sided polygon)Edit

 
 
 
 

12°: regular pentadecagon (15-sided polygon)Edit

 
 
 
 

15°: regular dodecagon (12-sided polygon)Edit

 
 
 
 

18°: regular decagon (10-sided polygon)[1]Edit

 
 
 
 

21°: sum 9° + 12°Edit

 
 
 
 

22.5°: regular octagonEdit

 
 
 
 , the silver ratio

24°: sum 12° + 12°Edit

 
 
 
 

27°: sum 12° + 15°Edit

 
 
 
 

30°: regular hexagonEdit

 
 
 
 

33°: sum 15° + 18°Edit

 
 
 
 

36°: regular pentagonEdit

[1]
 
  where φ is the golden ratio;
 
 

39°: sum 18° + 21°Edit

 
 
 
 

42°: sum 21° + 21°Edit

 
 
 
 

45°: squareEdit

 
 
 
 

54°: sum 27° + 27°Edit

 
 
 
 

60°: equilateral triangleEdit

 
 
 
 

67.5°: sum 7.5° + 60°Edit

 
 
 
 

72°: sum 36° + 36°Edit

 
 
 
 

75°: sum 30° + 45°Edit

 
 
 
 

90°: fundamentalEdit

 
 
 
 

List of trigonometric constants of 2π/nEdit

For cube roots of non-real numbers that appear in this table, one has to take the principal value, that is the cube root with the largest real part; this largest real part is always positive. Therefore, the sums of cube roots that appear in the table are all positive real numbers.

 

NotesEdit

Uses for constantsEdit

As an example of the use of these constants, consider the volume of a regular dodecahedron, where a is the length of an edge:

 

Using

 
 

this can be simplified to:

 

Derivation trianglesEdit

 
Regular polygon (n-sided) and its fundamental right triangle. Angles: a = 180°/n and b =90(1 − 2/n

The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructibility 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
    • 3 × 2n-sided regular polygons, for n = 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: icositetragon (24-sided)
      • 86.25°-3.75°-90° triangle: tetracontaoctagon (48-sided)
      • 88.125°-1.875°-90° triangle: enneacontahexagon (96-sided)
      • 89.0625°-0.9375°-90° triangle: 192-gon
      • 89.53125°-0.46875°-90° triangle: 384-gon
      • ...
    • 4 × 2n-sided
      • 45°-45°-90° triangle: square (4-sided)
      • 67.5°-22.5°-90° triangle: octagon (8-sided)
      • 78.75°-11.25°-90° triangle: hexadecagon (16-sided)
      • 84.375°-5.625°-90° triangle: triacontadigon (32-sided)
      • 87.1875°-2.8125°-90° triangle: hexacontatetragon (64-sided)
      • 88.09375°-1.40625°-90° triangle: 128-gon
      • 89.046875°-0.703125°-90° triangle: 256-gon
      • ...
    • 5 × 2n-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)
      • 88.875°-1.125°-90° triangle: 160-gon
      • 89.4375°-0.5625°-90° triangle: 320-gon
      • ...
    • 15 × 2n-sided
    • ...
There are also higher constructible regular polygons: 17, 51, 85, 255, 257, 353, 449, 641, 1409, 2547, ..., 65535, 65537, 69481, 73697, ..., 4294967295.)
  • 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 (9-sided)
      • 80°-10°-90° triangle: octadecagon (18-sided)
      • 85°-5°-90° triangle: triacontahexagon (36-sided)
      • 87.5°-2.5°-90° triangle: heptacontadigon (72-sided)
      • ...
    • 45 × 2n-sided
      • 86°-4°-90° triangle: tetracontapentagon (45-sided)
      • 88°-2°-90° triangle: enneacontagon (90-sided)
      • 89°-1°-90° triangle: 180-gon
      • 89.5°-0.5°-90° triangle: 360-gon
      • ...

Calculated trigonometric values for sine and cosineEdit

The trivial valuesEdit

In degree format, sin and cos of 0, 30, 45, 60, and 90 can be calculated from their right angled triangles, using the Pythagorean theorem.

In radian format, sin and cos of π / 2n can be expressed in radical format by recursively applying the following:

  and so on.
  and so on.

For example:

 
  and  
  and  
  and  
  and  
  and  

and so on.

Radical form, sin and cos of π/(3 × 2n)Edit

 
  and  
  and  
  and  
  and  
  and  
  and  

and so on.

Radical form, sin and cos of π/(5 × 2n)Edit

 
  ( Therefore   )
  and  
  and  
  and  
  and  
  and  

and so on.

Radical form, sin and cos of π/(5 × 3 × 2n)Edit

 
  and  
  and  
  and  
  and  
  and  

and so on.

Radical form, sin and cos of π/(17 × 2n)Edit

If   and   then

 

Therefore, applying induction:

 
  and  

Radical form, sin and cos of π/(257 × 2n) and π/(65537 × 2n)Edit

The induction above can be applied in the same way to all the remaining Fermat primes (F3=223+1=28+1=257 and F4=224+1=216+1=65537), the factors of π whose cos and sin radical expressions are known to exist but are very long to express here.

  and  
  and  

Radical form, sin and cos of π/(255 × 2n), π/(65535 × 2n) and π/(4294967295 × 2n)Edit

D = 232 - 1 = 4,294,967,295 is the largest odd integer denominator for which radical forms for sin(π/D) and cos (π/D) are known to exist.

Using the radical form values from the sections above, and applying cos(A-B) = cosA cosB + sinA sinB, followed by induction, we get -

  and  
  and  

Therefore, using the radical form values from the sections above, and applying cos(A-B) = cosA cosB + sinA sinB, followed by induction, we get -

  and  
  and  

Finally, using the radical form values from the sections above, and applying cos(A-B) = cosA cosB + sinA sinB, followed by induction, we get -

  and  
  and  

The radical form expansion of the above is very large, hence expressed in the simpler form above.

n × π/(5 × 2m)Edit

 
Chord(36°) = a/b = 1/φ, i.e., the reciprocal of the golden ratio, from Ptolemy's theorem

Geometrical methodEdit

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/φ of the golden ratio. crd is the chord function,

 

(See also Ptolemy's table of chords.)

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

 

so

 

Algebraic methodEdit

If θ is 18° or -54°, then 2θ and 3θ add up to 5θ = 90° or -270°, therefore sin 2θ is equal to cos 3θ.

 
So,  , which implies  

Therefore,

  and   and
  and  

Alternately, the multiple-angle formulas for functions of 5x, where x ∈ {18, 36, 54, 72, 90} and 5x ∈ {90, 180, 270, 360, 450}, 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 quartic equation can be solved as a quadratic in y2.
  • 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 × π/20Edit

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

n × π/30Edit

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/60Edit

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.

Strategies for simplifying expressionsEdit

Rationalizing the denominatorEdit

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.

Splitting a fraction in twoEdit

Sometimes it helps to split the fraction into the sum of two fractions and then simplify both separately.

Squaring and taking square rootsEdit

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.

Simplifying nested radical expressionsEdit

In general nested radicals cannot be reduced. But if

 

with a, b, and c rational, we have

 

is rational, then both

 

are rational; then we have

 

For example,

 
 

See alsoEdit

ReferencesEdit

  1. ^ a b Bradie, Brian (Sep 2002). "Exact values for the sine and cosine of multiples of 18°: A geometric approach". The College Mathematics Journal. 33 (4): 318–319. doi:10.2307/1559057. JSTOR 1559057.

External linksEdit