In mathematics, the values of the trigonometric functions can be expressed approximately, 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 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.
The trigonometric functions of angles that are multiples of 15°, 18°, or 22.5° have simple algebraic values.
[1] In the table below, the label "Undefined" represents a ratio
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, these entries take the value
For angles outside of this range, trigonometric values can be found by applying reflection and shift identities such as A trigonometric number is a number that can be expressed as the sine or cosine of a rational multiple of π radians.
[2] The minimal polynomials of trigonometric numbers can be explicitly enumerated.
[3] In contrast, by the Lindemann–Weierstrass theorem, the sine or cosine of any non-zero algebraic number is always transcendental.
[4] The real part of any root of unity is a trigonometric number.
By Niven's theorem, the only rational trigonometric numbers are 0, 1, −1, 1/2, and −1/2.
[5] 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.
is a constructible angle because 12 is a power of two (4) times a Fermat prime (3).
A geometric way of deriving the sine or cosine of 45° is by considering an isosceles right triangle with leg length 1.
The values of sine and cosine of 30 and 60 degrees are derived by analysis of the equilateral triangle.
Bisecting one corner, the special right triangle with angles 30-60-90 is obtained.
may be derived using the multiple angle formulas for sine and cosine.
[9] 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: The Pythagorean identity then gives
, and the double and triple angle formulas give sine and cosine of 36°, 54°, and 72°.
For example, 22.5° (π/8 rad) is half of 45°, so its sine and cosine are:[11] Repeated application of the half-angle formulas leads to nested radicals, specifically nested square roots of 2 of the form
In general, the sine and cosine of most angles of the form
can be expressed using nested square roots of 2 in terms of
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:[13][14] The sines and cosines of other constructible angles of the form
radians have trigonometric values that can be expressed with 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.
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.
[15][16] Alternatively, by De Moivre's formula: Taking cube roots and adding or subtracting the equations, we have:[16]