Trigonometric constants expressed in real radicals

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 term 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°, the values of sine, cosine, and tangent cannot be expressed in terms of real radicals. Note that 1° = π/180 radians.

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 either the angle is rational number of degrees, or the angle is 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 article

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. This article only gives the cases based on the Fermat primes 3 and 5. Thus for example cos(2π/17), given in the article Heptadecagon, is not given here.

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 angles

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° 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 angles

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°: fundamental

1.5°: regular hecatonicosagon (120-sided polygon)

1.875°: regular enneacontahexagon (96-sided polygon)

2.25°: regular octacontagon (80-sided polygon)

2.8125°: regular hexacontatetragon (64-sided polygon)

3°: regular hexacontagon (60-sided polygon)

3.75°: regular tetracontaoctagon (48-sided polygon)

4.5°: regular tetracontagon (40-sided polygon)

5.625°: regular triacontadigon (32-sided polygon)

6°: regular triacontagon (30-sided polygon)

7.5°: regular icositetragon (24-sided polygon)

9°: regular icosagon (20-sided polygon)

11.25°: regular hexadecagon (16-sided polygon)

12°: regular pentadecagon (15-sided polygon)

15°: regular dodecagon (12-sided polygon)

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

21°: sum 9° + 12°

22.5°: regular octagon

(the silver ratio)

24°: sum 12° + 12°

27°: sum 12° + 15°

30°: regular hexagon

33°: sum 15° + 18°

36°: regular pentagon

[1]
where φ is the golden ratio;

39°: sum 18° + 21°

42°: sum 21° + 21°

45°: square

54°: sum 27° + 27°

60°: equilateral triangle

67.5°: sum 7.5° + 60°

72°: sum 36° + 36°

75°: sum 30° + 45°

90°: fundamental

List of trigonometric constants to 2π/n

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

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.

(Higher constructible regular polygons don't make whole degree angles: 17, 51, 85, 255, 257, ..., 65537, ..., 4294967295. At the same time, neither do some of the above listed polygons.)

Calculated trigonometric values for sine and cosine

The trivial values

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)

and
and
and
and
and

and so on.

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

( Therefore )
and
and
and
and
and

and so on.

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

and
and
and
and

and so on.

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

If and then

Therefore, applying induction:

and

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

Note that 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)

D = 232 - 1 = 4,294,967,295 is the largest odd integer denominator for which radical form 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

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

n × π/(5 × 2m)

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

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/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 method

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:

One solution is zero, and the resulting quartic equation can be solved as a quadratic in y2.
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.

Strategies for simplifying expressions

Rationalizing 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.

Splitting a fraction in two

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

Squaring and taking square roots

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 expressions

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 also

References

  1. 1 2 Bradie, Brian (Sep 2002). "Exact values for the sine and cosine of multiples of 18°—A geometric approach". The College Mathematics Journal. 33: 318–319.

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