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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); 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 derivable in radicals 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°. Note that 1° = /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. == 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 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 given in the article 17-gon, 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 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 ''Generating trigonometric tables''. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Trigonometric constants expressed in real radicals」の詳細全文を読む スポンサード リンク
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