In mathematics, the values of the
trigonometric functions
In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in a ...
can be expressed approximately, as in
, or exactly, as in
. While
trigonometric tables contain many approximate values, the exact values for certain angles can be expressed by a combination of arithmetic operations and
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
s.
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 90°.
[Abramowitz, Milton and Irene A. Stegun, p. 74] For angles outside of this range, trigonometric values can be found by applying the
reflection and shift identities. In the table below,
stands for the ratio 1:0. These values can also be considered to be undefined (see
division by zero).
:
Expressibility with square roots
Some exact trigonometric values, such as
, can be expressed in terms of a combination of arithmetic operations and square roots. Such numbers are called
constructible, because one length can be
constructed by compass and straightedge from another if and only if the ratio between the two lengths is such a number.
However, some trigonometric values, such as
, have been proven to not be constructible.
The sine and cosine of an angle are constructible if and only if the angle is constructible. If an angle is a rational multiple of radians, whether or not it is constructible can be determined as follows. Let the angle be
radians, where ''a'' and ''b'' are
relatively prime integers. Then it is constructible if and only if the
prime factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are s ...
of the denominator, ''b'', consists of any number of
Fermat primes, each with an exponent of 1, times any
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negat ...
. For example,
and
are constructible because they are equivalent to
and
radians, respectively, and 12 is the product of 3 and 4, which are a Fermat prime and a power of two, and 15 is the product of Fermat primes 3 and 5. On the other hand,
is not constructible because it corresponds to a denominator of 9 = 3
2, and the Fermat prime cannot be raised to a power greater than one. As another example,
is not constructible, because the denominator of 7 is not a Fermat prime.
Derivations of constructible values
The values of trigonometric numbers can be derived through a combination of methods. The values of sine and cosine of 30, 45, and 60 degrees are derived by analysis of the
30-60-90 and 90-45-45 triangles. If the angle is expressed in radians as
, this takes care of the case where ''a'' is 1 and ''b'' is 2, 3, 4, or 6.
Half-angle formula
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 cosine half-angle formula leads to
nested square roots that continue in a pattern where each application adds a
to the numerator and the denominator is 2. For example:
:
:
Sine of 18°
Cases where the denominator, ''b'', is 5 times a power of 2 can start from the following derivation of
, since
radians. The derivation uses the
multiple angle formulas for sine and cosine. By the double angle formula for sine:
:
By the triple angle formula for cosine:
:
Since sin(36°) = cos(54°), we equate these two expressions and cancel a factor of cos(18°):
:
This quadratic equation has only one positive root:
:
Using other identities
The sines and cosines of many other angles can be derived using the results described above and a combination of the multiple angle formulas and the
sum and difference formulas. For example, for the case where ''b'' is 15 times a power of 2, since
, its cosine can be derived by the cosine difference formula:
:
Denominator of 17
Since 17 is a Fermat prime, a regular
17-gon is constructible, which means that the sines and cosines of angles such as
radians can be expressed in terms of square roots. In particular, in 1796,
Carl Friedrich Gauss showed that:
[Arthur Jones, Sidney A. Morris, Kenneth R. Pearson, ''Abstract Algebra and Famous Impossibilities'', Springer, 1991, ]
p. 178.
/ref>[Callagy, James J. "The central angle of the regular 17-gon", ''Mathematical Gazette'' 67, December 1983, 290–292.]
:
The sines and cosines of other constructible angles with a denominator divisible by 17 can be derived from this one.
Roots of unity
An irrational number that can be expressed as the sine or cosine of a rational multiple of radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
s is called a ''trigonometric number''.[Niven, Ivan. ''Numbers: Rational and Irrational'', 1961. Random House. New Mathematical Library, Vol. 1. .] Since the case of a sine can be omitted from this definition. Therefore any trigonometric number can be written as , where ''k'' and ''n'' are integers. This number can be thought of as the real part of the complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
. De Moivre's formula shows that numbers of this form are roots of unity:
:
Since the root of unity is a root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of the polynomial ''x''''n'' − 1, it is algebraic
Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings.
Algebraic may also refer to:
* Algebraic data type, a data ...
. 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 real part of any root of unity is trigonometric, unless it is rational. By Niven's theorem, the only rational numbers that can be expressed as the real part of a root of unity are 0, 1, −1, 1/2, and −1/2.
Extended table of exact values
See also
* List of trigonometric identities
References
Bibliography
*
{{Irrational number
Irrational numbers
Trigonometry