In mathematics, the values of the trigonometric functions. While trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and square roots. The angles with trigonometric values that are expressible in this way are exactly those that can be constructed with a compass and straight edge, and the values are called constructible numbers.

Common angles

The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values. These values are listed in the following table for angles from 0° to 45° (see below If the codomain of the trigonometric functions is taken to be the real numbers these entries are undefined, whereas if the codomain is taken to be the projectively extended real numbers (see division by zero).For angles outside of this range, trigonometric values can be found by applying reflection and shift identities, such as

For example, for any greater than , to find its sine, follow these steps to instead operate within the range of 0 to :

1. While it is greater than 2, subtract 2 from it. Now try to find .

2. If it is greater than , subtract from it and try to find .

3. If it is greater than , subtract from it and try to find .

4. If it is greater than , subtract it from . If you are trying to find cosine due to step 3, now try to find . If you are trying to find sine because you did not enter step 3, try to find .

Trigonometric numbers

A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of radians., where k and n are integers. This number can be thought of as the real part of the complex number. De Moivre's formula shows that numbers of this form are roots of unity:

(Conversely, the real part of any root of unity is a trigonometric number.) Since the root of unity is a root of the polynomial xn−1, it is algebraic. Since the trigonometric number is the average of the root of unity and its complex conjugate, and algebraic numbers are closed under arithmetic operations, every trigonometric number is algebraic. The minimal polynomials of trigonometric numbers can be explicitly enumerated. In contrast, by the Lindemann–Weierstrass theorem, the sine or cosine of any non-zero algebraic number is always transcendental.

By Niven's theorem, the only rational trigonometric numbers are 0, 1, −1, 1/2, and −1/2.

Constructibility

An angle can be constructed with a compass and straightedge if and only if its sine (or equivalently cosine) can be expressed by a combination of arithmetic operations and square roots applied to integers. radians, where a and b are relatively prime integers, the prime factorization of the denominator, b, is the product of some power of two and any number of distinct Fermat primes (a Fermat prime is a prime number one greater than a power of two).

is not the product of distinct, since 7 is not a Fermat prime.

It results from the above characterisation that an angle of an integer number of degrees is constructible if and only if this number of degrees is a multiple of .

Constructible values

45°

From a reflection identity. Substituting into the Pythagorean trigonometric identity, one obtains the minimal polynomial.

, giving the same value for the sine or cosine of 45° given above.

30° and 60°

The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle. In an equilateral triangle, the 3 angles are equal and sum to 180°, therefore each corner angle is 60°. Bisecting one corner, the special right triangle.

18°, 36°, 54°, and 72°

may be derived using the multiple angle formulas for sine and cosine.

, where is the golden ratio.

A geometric method of finding the sines of these angles comes from analyzing a regular pentagon. Two of its diagonals form an angle of 36° that can be inscribed in a circle, and one side of the pentagon forms a chord of the circle that subtends this inscribed angle. To compute the chord's length, form a quadrilateral from four points of the pentagon. If the diagonals each have a length of one, then applying Ptolemy's theorem, where x is the chord length. The positive root of this equation is this value:

Bisecting the 36° angle then creates a right triangle with angles of 18° and 72°, and cuts the chord in half, giving the same value for the sine of 18° given above. The sine value of 72° then follows from the Pythagorean theorem, and the sine values for 36° and 54° follow from sin(18°) and the double- and triple-angle formulas.

Remaining multiples of 3°

The sines and cosines of all other angles between 0 and 90° that are multiples of 3° can be derived from the angles described above and the sum and difference formulas. Specifically,

Half angles

If the denominator, b, is multiplied by additional factors of 2, the sine and cosine can be derived with the half-angle formulas. For example, 22.5° (/8 rad) is half of 45°, so its sine and cosine are:

Repeated application of the half-angle formulas leads to nested radicals, specifically nested square roots of 2. Specifically, if one can write an angle as , then then

Denominator of 17

Since 17 is a Fermat prime, a regular 17-gon radians can be expressed in terms of square roots. In particular, in 1796, Carl Friedrich Gauss showed that:

radians (for integers and ) can be derived from this one.

Non-constructibility of 1°

As discussed in cannot be expressed using only square roots. A related question is whether it can be expressed using cube roots. The following two approaches can be used, but both result in an expression that involves the cube root of a complex number. Galois theory shows that the casus irreducibilis cannot be expressed in terms of solely real radicals.

is constructible, an expression for it could be plugged into Cardano's formula. However, since all three roots of the cubic are real, this is an instance of casus irreducibilis, and the expression would require taking the cube root of a complex number.

Alternatively, by De Moivre's formula:

Taking cube roots and adding or subtracting the equations, we have:

See also

• Ailles rectangle, used to find exact values for multiples of 15°

• List of trigonometric identities

References

Bibliography